Central $L$-values of newforms and local polynomials

TL;DR

This paper characterizes the vanishing of twisted central L-values for newforms of square-free level using local polynomials and Hecke operators, connecting these to harmonic Maass forms and generalized Hurwitz class numbers.

Contribution

It introduces a new criterion for vanishing of twisted central L-values based on explicit polynomials and extends existing results to higher weights and levels.

Findings

Vanishing of L-values characterized by constant polynomials.

Established identity between constants and Hurwitz class numbers.

Numerical examples provided for specific weights and levels.

Abstract

In this paper, we characterize the vanishing of twisted central $L$-values attached to newforms of square-free level in terms of so-called local polynomials and the action of finitely many Hecke operators thereon. Such polynomials are the ``local part'' of certain locally harmonic Maass forms constructed by Bringmann, Kane and Kohnen in $2015$. We offer a second perspective on this characterization for weights greater than $4$ by adapting results of Zagier to higher level. To be more precise, we establish that a twisted central $L$-value attached to a newform vanishes if and only if a certain explicitly computable polynomial is constant. We conclude by proving an identity between these constants and generalized Hurwitz class numbers, which were introduced by Pei and Wang in $2003$. We provide numerical examples in weight $4$ and levels $7$, $15$, $22$, and offer some questions for future work.