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

Abstract

Let $F$ be a self-dual Hecke-Maa\ss\ form for ${\rm{GL}}(3)$ underlying the symmetric square lift of a ${\rm{GL}}(2)$-newform of square-free level and trivial nebentypus. In this paper, we are interested in the first moments of the central values of ${\rm{GL}}(3) \times {\rm{GL}}(2)$ $L$-functions and ${\rm{GL}}(2)$ $L$-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 $f\in \mathcal{B}^\ast_k(M)$ 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 $M^{13/64+\varepsilon }\le q \le M^{11/40-\varepsilon}$ and $M> q^\delta$ for any $\varepsilon, \delta>0$ 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 Lindel\"of 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 $O(K^{-1/4+\varepsilon})$, respectively.