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 $\mathrm{GL}(n)\times \mathrm{GL}(n)$ 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.

arXiv comment44 pages, to appear in Forum math, Sigma

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• The subconvexity problem for $\GL_{2}$
• The orbit method and analysis of automorphic forms
• Upper bounds on L-functions at the edge of the critical strip
• Applications of analytic newvectors for $\mathrm{GL}(n)$
• A hybrid asymptotic formula for the second moment of Rankin-Selberg L-functions
• The sixth moment of automorphic $L$-functions
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Cited by in corpus (1)

• Zeros of Rankin-Selberg $L$-functions in families