The second moment of $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions

TL;DR

This paper derives an asymptotic formula for the second moment of central values of $ ext{GL}(n) imes ext{GL}(n)$ Rankin--Selberg $L$-functions over automorphic representations with growing conductors, using advanced integral and representation techniques.

Contribution

It provides the first asymptotic expansion for the second moment of these $L$-functions, employing integral representations, Eisenstein series, and invariance of newvectors.

Findings

Asymptotic expansion of the second moment established

Techniques involve integral representations and regularized Eisenstein series

Results advance understanding of $L$-function value distributions

Abstract

We prove an asymptotic expansion of the second moment of the central values of the $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions $L(1/2,\pi\otimes\pi_0)$, for a fixed cuspidal automorphic representation $\pi_0$, over the family of $\pi$ with analytic conductors bounded by a quantity which is tending off to infinity. Our proof uses the integral representations of the $L$-functions, period with regularized Eisenstein series, and the invariance properties of the analytic newvectors.