First moment of Rankin-Selberg central L-values and subconvexity in the level aspect

Abstract: Let $1\le N<M$ with $N$ and $M$ coprime and square-free. Through classical analytic methods we estimate the first moment of central $L$-values $ L(1/2,f\times g) $ where $f\in S^*_k(N)$ runs over primitive holomorphic forms of level $N$ and trivial nebentypus and $g$ is a given form of level $M$. As a result, we recover the bound $ L(1/2,f\times g) \ll_\varepsilon (N + \sqrt{M}) N^\varepsilon M^\varepsilon $ when $g$ is dihedral. The first moment method also applies to the special derivative $L'(1/2,f\times g)$ under the assumption that it is non-negative for all $f\in S^*_k(N)$.