ORDER
Janice and Christopher Marten sued two Indiana state employees under 42 U.S.C. *447§ 1983 for alleged civil-rights violations in connection with a state tax investigation. Specifically, the Martens claimed that the two officials — Rick Albrecht, a tax auditor, and Andrew Swain, an attorney in the Office of the State Attorney General— fraudulently procured a search warrant for their home and business, searched and seized items beyond the scope of that warrant, used administrative subpoenas in furtherance of a criminal investigation, and withheld exculpatory evidence. The district court dismissed the § 1983 claim as time-barred — a decision that, as both sides now acknowledge, runs contrary to our holding in Julian v. Hanna, 732 F.3d 842 (7th Cir.2013). Even so, the defendants argue that we can affirm the dismissal because the complaint fails to plead adequate facts to state a claim for relief. But they did not make this argument in the district court, so we cannot rely on it as an alternative ground to support affirmance. Swain also argues that he is entitled to absolute prosecutorial immunity. This argument was raised below but is not appropriate for resolution on the pleadings. We vacate and remand.
I. Background
Because the case was dismissed at the pleading stage, we accept the facts alleged in the Martens’ complaint as true. Serino v. Hensley, 735 F.3d 588, 590 (7th Cir.2013). In 2007 the Indiana Department of Revenue (“the DOR”) initiated a tax audit of Janice Marten’s jewelry business. Al-brecht was the original auditor on the case. From July 2007 to January 2008, Janice asked Albrecht for status updates on the audit, but he did not respond. In 2008 the DOR initiated civil proceedings against Janice and her business, alleging that they had failed to collect and remit the correct sales tax.
The Office of the Indiana Attorney General dispatched Swain, chief counsel in its Tax Litigation Section, to represent the DOR in the civil proceedings. In June 2008 Swain subpoenaed and deposed Janice and her husband, Christopher. The couple asked Swain if they needed a lawyer. He told them that they did not. What Swain did not tell them was that Albrecht had already launched a criminal investigation. Before the depositions took place, Albrecht submitted an affidavit in support of a search warrant for the Martens’ home and the jewelry business. The Martens allege that the affidavit contained “misleading, if not false, statements of fact.”
Swain and other unidentified officials executed the search warrants on July 20, 2008. The search extended to items and locations not covered by the warrants. Afterward, Swain gave the Martens search inventories purporting to itemize the seized property, but the inventories omitted a substantial number of items.
While the search was ongoing, Albrecht handed Janice copies of tax assessments against her business. Two days later, the DOR filed formal Jeopardy Assessment Notices seeking about $900,000 from the jewelry business and the Martens in their capacity as officers of the business. The Martens retained counsel, and the lawyers met with the DOR Commissioner on October 10 and negotiated a settlement. The terms of the settlement called for the Martens to make a preliminary escrow deposit of $100,000 pending a review of the audit and a final determination of back taxes due. On October 15 the jewelry business delivered a check for $100,000 to the DOR.
Eight days later, the Martens were arrested on 14 criminal counts, including failure to pay taxes, tax evasion, and falsification of tax records. The charging information was based in part on Albrecht’s *448affidavit and evidence gleaned from the Martens’ depositions from the civil proceedings. The criminal charges remained pending for nearly two and a half years. Finally, on March 7, 2011, the state court dismissed all charges against the Martens after finding that the audit report had been purposefully withheld from them and their counsel.
In February 2012 the Martens filed suit in district court against Swain, Albrecht, the DOR, the Office of the Attorney General, and unknown representatives of these state offices. Count I alleged a § 1983 claim for violation of the Martens’ rights under the Fourth, Fifth, Sixth, and Fourteenth Amendments. Counts II, III, IV, and V alleged claims under Indiana law for malicious prosecution, intentional infliction of emotional distress, abuse of process, and conversion.
The defendants moved to dismiss the action in its entirety. The district court dismissed Counts I, III, IV, and V with prejudice, finding each count barred by the applicable statute of limitations. The court dismissed Count II (the state-law malicious-prosecution claim) without prejudice, and allowed the Martens to amend their complaint to address notice and immunity issues under the Indiana Tort Claims Act.
The amended complaint alleged a single state-law claim for malicious prosecution against Swain, Albrecht, and the State of Indiana. The defendants again moved to dismiss, this time for lack of subject-matter jurisdiction, or alternatively, based on the immunity provision of the Tort Claims Act. The Martens conceded that with the dismissal of the § 1983 claim, jurisdiction was now lacking. They asked the court to “remand” the case to state court.
The district court declined the remand request because the Martens had initiated the suit in federal court. And although no federal claim had survived the initial motion to dismiss, the court retained jurisdiction and addressed the state-law malicious-prosecution claim on. the merits. The court held that Swain and Albrecht were entitled to immunity under the Tort Claims Act and the State of Indiana had not consented to suit. Accordingly, the court dismissed the amended complaint with prejudice. This appeal followed.
II. Discussion
The Martens’ sole argument on appeal is that the district court erred in dismissing their § 1983 due-process claim for malicious prosecution as time-barred under the two-year statute of limitations.1 The judge concluded that the claim accrued in 2008 when Swain and Albrecht took the actions in question, rather than in 2011 when the criminal charges against the Martens were dismissed. Because the Martens filed suit in late 2012, the judge dismissed the claim as untimely.
That decision is incompatible with Julian v. Hanna, 732 F.3d 842, as Swain and Albrecht now admit. We held in Julian that a § 1983 claim for malicious prosecution “does not accrue until the criminal proceeding that gave rise to it ends in the claimant’s favor.” Id. at 845. Under this rule of accrual, the due-process claim accrued in 2011 when the criminal charges against the Martens were dismissed. The § 1983 claim — filed the following year— was therefore timely.
*449Swain and Albrecht concede the error but insist that we can affirm anyway because the Martens have failed to state a claim upon which relief can be granted. They identify two flaws in the complaint. First, they argue that it is factually insufficient because it does not specifically identify the misrepresentations Swain and Albrecht made in furtherance of the prosecution and does not allege that they made the misrepresentations with knowledge of their falsity. Second, they assert that the claim of malicious prosecution is not cognizable as a deprivation of due process.
We may affirm on any ground fairly supported by the record if the appellee preserved the argument below. Burns v. Orthotek, Inc. Emps.’ Pension Plan & Trust, 657 F.3d 571, 575 (7th Cir.2011) (explaining that “[w]e can affirm on any ground that the record fairly supports and the appellee has not waived” (internal quotation marks omitted)). “Only if a party raises an argument both here and in the district court may we use it as an alternate means to affirm.... ” Tully v. Barada, 599 F.3d 591, 594 (7th Cir.2010).
Swain and Albrecht did not attack the factual sufficiency of the § 1983 malicious-prosecution claim in the district court. Their motion to dismiss did not address either the false-statement or knowledge components of the claim — -the very allegations they now contend are insufficiently developed. Because they did not raise the argument below, it lies beyond our reach as an alternative ground for affirmance.
Swain and Albrecht did dispute the Martens’ framing of their malicious-prosecution claim as a Fourteenth Amendment due-process violation, so that argument was adequately preserved. But it’s not a winner, at least not at this early stage of the case. Julian authorized a due-process claim for malicious prosecution in Indiana because state immunity law bars the common-law cause of action, making state remedies inadequate. 732 F.3d at 848-49. Our decision in Serino fleshed out some of the contours of the claim. 735 F.3d at 593-94. Both cases postdate the dismissal of this suit, and because the due-process claim was erroneously dismissed as time-barred, the Martens have not yet had an opportunity to address these new legal developments. Now that we’ve corrected the statute-of-limitations error,' they are entitled to try to plead sufficient facts to support a due-process claim as understood in Julian and Serino.
Finally, Swain asks us to affirm the dismissal of the § 1983 claim against him because his position as prosecutor furnishes absolute immunity. The Supreme Court has characterized prosecutorial immunity from suit under § 1983 as “absolute.” Van de Kamp v. Goldstein, 555 U.S. 335, 341-42, 129 S.Ct. 855, 172 L.Ed.2d 706 (2009) (noting that “[t]he public trust of the prosecutor’s office would suffer were the prosecutor to have in mind his own potential damages liability when making prosecutorial decisions” (internal quotation marks omitted)). But this “immunity may not apply when a prosecutor is not acting as an officer of the court, but is • instead engaged in other tasks, say, investigative or administrative tasks.” Id. at 342, 129 S.Ct. 855 (internal quotation marks omitted). Thus, it’s “the nature of the function performed, not the identity of the actor who performed it, that inform[s] our immunity analysis.” Forrester v. White, 484 U.S. 219, 229, 108 S.Ct. 538, 98 L.Ed.2d 555 (1988).
We are hesitant to engage in this functional analysis at the pleading stage, particularly where the district court has yet to pass on the issue. See Alvarado v. Litscher, 267 F.3d 648, 651 (7th Cir.2001) (“Because an immunity defense usually depends on the facts of the ease, dismissal at *450the pleading stage is inappropriate.... ”). The Martens were not required to anticipate and plead around the affirmative-defense of prosecutorial immunity, and their complaint makes no implicit concession that would “plead [them] out of court by alleging (and thus admitting) the ingredients” of such a defense. U. S. Gypsum Co. v. Ind. Gas Co., 350 F.3d 623, 626 (7th Cir.2003). Since “we have not sharply defined the boundary between prosecutorial and investigatory actions,” Alexander v. Shan, 161 Fed.Appx. 571, 575 (7th Cir.2005), any immunity determination involves a necessarily fact-specific line-drawing exercise that would benefit from greater development of the record below. On the complaint alone, we cannot conclude with any certainty whether and to what extent Swain is entitled to immunity.
Accordingly, we VACATE the judgment dismissing the § 1983 claim for malicious prosecution and REMAND for further proceedings consistent with this opinion.
. The Martens do not seek to revive any other claim, nor do they challenge the dismissal of all claims against the State of Indiana, the DOR, the Office of the Attorney General, and unnamed representatives of these state offices. Accordingly, our review is limited to the § 1983 due-process claim for malicious prosecution against Swain and Albrecht.