A bound for twists of $\rm GL_3\times GL_2$ $L$-functions with composite modulus

Abstract

Let $\pi$ be a Hecke-Maass cusp form for $\rm SL_3(\mathbf{Z})$ and let $g$ be a holomorphic or Maass cusp form for $\rm SL_2(\mathbf{Z})$. Let $\chi$ be a primitive Dirichlet character of modulus $M=M_1M_2$ with $M_i$ prime, $i=1,2$. Suppose that $M^{1/2+2\eta}<M_1<M^{1-2\eta}$ with $0<\eta<1/8$. Then we have $$ L\left(\frac{1}{2},\pi\otimes g \otimes \chi\right)\ll_{\pi,g,\varepsilon} M^{3/2-\eta+\varepsilon}. $$