𝐿-values of harmonic Maass forms

Abstract

Bruinier, Funke, and Imamoglu have proved a formula for what can philosophically be called the “central L L -value” of the modular j j -invariant. Previously, this had been heuristically suggested by Zagier. Here, we interpret this “ L L -value” as the value of an actual L L -series, and extend it to all integral arguments and to a large class of harmonic Maass forms which includes all weakly holomorphic cusp forms. The context and relation to previously defined L L -series for weakly holomorphic and harmonic Maass forms are discussed. These formulas suggest possible arithmetic or geometric meaning of L L -values in these situations. The key ingredient of the proof is to apply a recent theory of Lee, Raji, and the authors to describe harmonic Maass L L -functions using test functions.