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

Let F$F$ be a self‐dual Hecke–Maaß form for GL(3)${\mathrm{GL}}(3)$ underlying the symmetric square lift of a GL(2)${\mathrm{GL}}(2)$‐newform of square‐free level and trivial nebentypus. In this paper, we are interested in the first moments of the central values of GL(3)×GL(2)$\rm GL(3)\times GL(2)$ L$L$‐functions and GL(2)$\rm GL(2)$ L$L$‐functions. As a result, we obtain an estimate for the first moment for L(1/2,F⊗f)$L(1/2, F\otimes f)$ in a family, where F$F$ is of the level q2$q^2$, and f∈Bk*(M)$f\in \mathcal {B}^\ast _k(M)$ for any primes q,M≥2$q,M\ge 2$ such that (q,M)=1$(q,M)=1$. We prove the subconvex bound for L(1/2,F⊗f)$L(1/2, F\otimes f)$ involving the level aspects simultaneously in the range M13/64+ε≤q≤M11/40−ε$M^{13/64+\varepsilon }\le q \le M^{11/40-\varepsilon }$ and M>qδ$M> q^\delta$ for any ε,δ>0$\varepsilon, \delta >0$ for the first time. Moreover, we further investigate the first moments of these L$L$‐functions in the weight k$k$ aspect over K≤k≤2K$K\le k\le 2K$, with K$K$ 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⊗f)$L(1/2, f)L(1/2, F\otimes f)$ of degree 8 and an asymptotic formula for the first moment of L(1/2,F⊗f)$L(1/2, F\otimes f)$ with an error term of O(K−1/4+ε)$O(K^{-1/4+\varepsilon })$, respectively.