Simultaneous non-vanishing of twists

Let $f$ be a newform of even weight $k$, level $M$ and character $\psi$ and let $g$ be a newform of even weight $l$, level $N$ and character $\eta$. We give a generalization of a theorem of Elliott, regarding the average values of Dirichlet $L$-functions, in the context of twisted modular $L$-functions associated to $f$ and $g$. Using this result, we find a lower bound in terms of $Q$ for the number of primitive Dirichlet characters modulo prime $q\leq Q$ whose twisted product $L$-functions $L_{f,\chi}(s_0) L_{g,\chi}(s_0)$ are non-vanishing at a fixed point $s_0=\sigma_0+it_0$ with $\frac{1}{2}<\sigma_0\leq 1$.