The second moment of Dirichlet twists of a $\textrm{GL}_{4}$ automorphic $L$-function

TL;DR

This paper derives an asymptotic formula for the second moment of Dirichlet twists of a GL(4) automorphic L-function on the critical line, averaged over characters and conductors, with bounds on the error term.

Contribution

It provides the first asymptotic formula for the second moment of these twists for GL(4) automorphic L-functions, including hybrid error bounds.

Findings

Asymptotic formula for the second moment derived

Hybrid bounds for the error term established

Results applicable to automorphic representations with unitary central character

Abstract

In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic $L$-function $L(s, \pi)$ on the critical line averaged over characters and conductors, where $\pi$ denotes an irreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$ with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.