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

Abstract

We prove an asymptotic expansion of the second moment of the central values of the GL(n) × GL(n) Rankin-Selberg L-functions L(1/2, π ⊗ π 0 ), for a fixed cuspidal automorphic representation π 0 , over the family of π 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.

Hecke-Maass cusp forms on PGL_n of large levels with non-vanishing L-values

“…An advantage of our method is that one can deal with arbitrary GL n uniformly by a relatively soft method except that the exact evaluation of local zeta-integrals is required to compute the main term. We should mention recent researches [7], [29] and [54] done in a similar spirit to our work dealing with analytic problems related to automorphic forms on GL(n) by a soft method.…”