First moments of $\mathrm{GL}(3)\times \mathrm{GL}(2)$ and $\mathrm{GL}(2)$ $L$-functions and their applications

Abstract

Let $F$ be a self-dual Hecke-Maaß form for $\mathrm{GL}(3)$ underlying the symmetric square lift of a $\mathrm{GL}(2)$-newform of square-free level and trivial nebentypus. In this paper, we are interested in the first moments of the central values of $\mathrm{GL}(3)\times \mathrm{GL}(2)$ $L$-functions and $\mathrm{GL}(2)$ $L$-functions. As a result, we obtain an estimate for the first moment for $L(1/2,F\otimes f)$ over a family, where $F$ is of the level $q^2$, and $f\in {\mathcal{B}}_k^{\ast }(M)$ for any primes $q,M\ge 2$ such that $(q,M)=1$. We prove the subconvex bound for $L(1/2,F\otimes f)$ involving the levels aspects simultaneously in the range $M^{13/64+\epsilon }\le q\le M^{11/40-\epsilon }$ and $M>q^{\delta }$ for any $\epsilon ,\delta >0$ for the first time. Moreover, we further investigate the first moments of these $L$-functions in the weight $k$ aspect over $K\le k\le 2K$, with $K$ being a large number. As the results, we obtain a Lindelöf average bound for the first moment of $L(1/2,f)L(1/2,F\otimes f)$ of degree 8 and an asymptotic formula for the first moment of $L(1/2,F\otimes f)$ with an error term of $O(K^{-1/4+\epsilon })$, respectively.