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

We prove an asymptotic expansion of the second moment of the central values of the $\mathrm{G}\mathrm{L}(n)\times \mathrm{G}\mathrm{L}(n)$ Rankin–Selberg $L$-functions $L(1\mathrm{/}2,\pi \otimes {\pi }_0)$, 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.