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 that is tending to infinity. Our proof uses the integral representations of the L-functions, period with regularised Eisenstein series and the invariance properties of the analytic newvectors.

In this paper, we study the extreme values of the Rankin–Selberg L-functions associated with holomorphic cusp forms in the vertical direction. Assuming the generalised Riemann hypothesis (GRH), we prove that

$$ \begin{align*} \underset{T^{\delta}\leq t\leq T}{\max}\bigg\lvert L\bigg(\frac12+it,f\times f\bigg)\bigg\rvert \geq\exp\bigg(C\sqrt{\frac{\log T\log\log\log T}{\log\log T}}\bigg) \end{align*} $$

with $C\leq \mathscr {X}\sqrt {1-\delta }$ , where $\mathscr {X}:=({2}/{\pi })\int _{0}^{\pi /3}\sin ^2\xi \,d\xi $ and $0\leq \delta <1$ .