On Hecke $L$-functions attached to half-integral weight modular forms

TL;DR

This paper studies the non-vanishing behavior of Hecke L-functions associated with half-integral weight modular forms on the real line, providing insights into their analytic properties.

Contribution

It offers new results on the non-vanishing of L-functions for half-integral weight modular forms, a less-explored area in analytic number theory.

Findings

Established non-vanishing results on the real line for these L-functions

Extended understanding of the analytic behavior of half-integral weight modular forms

Provided conditions under which the L-functions do not vanish

Abstract

We investigate non-vanishing properties of $L(f,s)$ on the real line, when $f$ is a Hecke eigenform of half-integral weight $k+{1\over 2}$ on $\Gamma_0(4).$