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 $F(s,\hat{I}\pm )$ of an $L$-function $F$ are well-understood. For example, it has poles when the positive number $\hat{I}\pm$ 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 $\mathbf{F}(\mathbf{s},\hat{I}\pm )$. 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 $\mathbf{F}(\mathbf{s},\hat{I}\pm )$. There are also significant differences; for instance, in the structure of the singularities.