A bound for twists of $\mathrm{G}{\mathrm{L}}_3\times \mathrm{G}{\mathrm{L}}_2$ $L$-functions with composite modulus

Abstract

Let $\ddot{I}€$ be a Hecke-Maass cusp form for $\mathrm{S}{\mathrm{L}}_3(\mathbf{Z})$ and let $g$ be a holomorphic or Maass cusp form for $\mathrm{S}{\mathrm{L}}_2(\mathbf{Z})$. Let $\ddot{I}‡$ be a primitive Dirichlet character of modulus $M=M_1{M}_2$ with $M_i$ prime, $i=1,2$. Suppose that $M^{1/2+2\hat{I}\cdot }<M_1<M^{1-2\hat{I}\cdot }$ with $0<\hat{I}\cdot <1/8$. Then we have $$L\left(\frac{1}{2},\ddot{I}€\otimes g\otimes \ddot{I}‡\right){\ll }_{\ddot{I}€,g,\epsilon }{M}^{3/2-\hat{I}\cdot +\epsilon }.$$

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