Multiple standard twists of $L$-functions

Abstract

The standard twist of $L$-functions plays a fundamental role in the Selberg class theory. It is defined as an absolutely convergent Dirichlet series and admits meromorphic continuation beyond the half-plane of absolute convergence. Nowadays, the analytic properties of the standard twist $F(s,\alpha)$ of an $L$-function $F$ are well-understood. For example, it has poles when the positive number $\alpha$ belongs to the so-called spectrum of $F$, and is entire otherwise. In this paper, for a given set ${\mathbf F}=\{F_1,\dots,F_N\}$ of $L$-functions and ${\mathbf s}\in{\mathbb C}^N$, we consider the multiple standard twist ${\mathbf F}({\mathbf s},\alpha)$. This is defined initially on a certain half-space of ${\mathbb C}^N$, and we describe its meromorphic continuation to the whole space. Results in the multidimensional case are, in many ways, analogous to those in the one-dimensional case. In particular, the spectrum of a multiple standard twist is relevant to the description of the set of poles of ${\mathbf F}({\mathbf s},\alpha)$. There are also significant differences; for instance, in the structure of the singularities.