The standard twist of L-functions revisited

TL;DR

This paper investigates the analytic properties of the standard twist of L-functions, revealing a new type of functional equation and detailed pole structure, which enhances understanding of the Selberg class.

Contribution

It introduces a novel functional equation for the standard twist of L-functions and characterizes its pole structure in detail.

Findings

F(s,α) satisfies a new type of functional equation resembling the Hurwitz-Lerch zeta function.

The polar structure of F(s,α) is characterized, identifying conditions for finitely or infinitely many poles.

A formula for the residues at poles of F(s,α) is provided.

Abstract

The analytic properties of the standard twist $F(s,\alpha)$, where $F(s)$ belongs to a wide class of $L$-functions, are of prime importance in describing the structure of the Selberg class. In this paper we present a deeper study of such properties. In particular, we show that $F(s,\alpha)$ satisfies a functional equation of a new type, somewhat resembling that of the Hurwitz-Lerch zeta function. Moreover, we detect the finer polar structure of $F(s,\alpha)$, characterizing in two different ways the occurrence of finitely or infinitely many poles as well as giving a formula for their residues.