On the standard twist of the L-functions of half-integral weight cusp forms

TL;DR

This paper investigates the analytic properties of the standard twist of L-functions associated with half-integral weight cusp forms, revealing a functional equation similar to Riemann's and analyzing zero distribution.

Contribution

It establishes a functional equation for the standard twist of these L-functions, resembling a degree 2 Riemann-type functional equation, and studies their growth and zero distribution.

Findings

The standard twist satisfies a Riemann-type functional equation.

The shape of the functional equation is akin to a degree 2 Hurwitz-Lerch equation.

Results on growth and zero distribution of the twisted L-functions.

Abstract

The standard twist $F(s,\alpha)$ of $L$-functions $F(s)$ in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where $F(s)$ satisfies a functional equation with the same $\Gamma$-factor of the $L$-functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such $L$-functions. We show that the standard twist $F(s,\alpha)$ satisfies a functional equation reflecting $s$ to $1-s$, whose shape is not far from a Riemann-type functional equation of degree 2 and may be regarded as a degree 2 analog of the Hurwitz-Lerch functional equation. We also deduce some result on the growth on vertical strips and on the distribution of zeros of $F(s,\alpha)$.