The second moment of $\mathrm{GL}(2)\times \mathrm{GL}(2)$ Rankin-Selberg $L$-functions in the level aspect

Abstract

We prove an asymptotic formula with a power-saving error term for a specific weighted second moment of $\mathrm{GL}(2)\times \mathrm{GL}(2)$ Rankin-Selberg $L$-function, $L(1/2,\pi\otimes \pi_0)$ over any number field $F$ where $\pi$ runs over representations with the non-archimedean conductor dividing an ideal which tends to infinity and $\pi_0$ is a fixed cuspidal representation unramified everywhere. The error term shows the square root cancellation under the assumption of the Generalised Ramanujan Conjecture.