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.
The standard twist of L-functions revisited
A. PERELLI
2021-01-01
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.
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