  In this paper we work over $\Gamma_0(N)$, for any $N$ and write the spectral
moment of a product of two distinct Rankin-Selberg convolutions at a general
point on the critical line $\frac{1}{2}+it$ as a main term plus a sharp error
term in the $t$ aspect and the spectral aspect. As a result we obtain hybrid
Weyl type subconvexity results in the $t$ and spectral aspects. Also, for fixed
modular forms $f$, $g$ of even weight $k\geq 4$ we show there exists a Maass
cusp form $u_j$ such that $L(1/2, f\times u_j)$, $L(1/2, g\times u_j)$ are
simultaneous non-zero.
