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In this note, we prove that given a Hecke-Maass cusp form $f$ for$SL_2(\mathbb{Z})$ and a sufficiently large integer $q=q_1q_2$ with $q_j\asymp\sqrt{q}$ being prime numbers for $j=1,2$, there exists a primitive Dirichletcharacter $\chi$ of conductor $q$ such that $L\left(\frac{1}{2},f\otimes\chi\right)L\left(\frac{1}{2},\chi\right)\neq 0$. To prove this, we establishasymptotic formulas of $L\left(\frac{1}{2},f\otimes\chi\right)L\left(\frac{1}{2},\chi\right)$ over the family of even primitiveDirichlet characters $\chi$ of conductor $q$ for more general $q$.

A note on simultaneous nonvanishing of Dirichlet $L$-functions and twists of Hecke-Maass $L$-functions