Hybrid bounds for twists of $GL(3)$ $L$-functions

Abstract

Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb{Z})$ and $\chi=\chi_1\chi_2$ a Dirichlet character with $\chi_i$ primitive modulo $M_i$. Suppose that $M_1$, $M_2$ are primes such that $\max\{\!(M|t|)^{\!1/3+2\delta/3\!},M^{2/5}|t|^{-9/20\!}, M^{1/2+2\delta}|t|^{-3/4+2\delta}\}(M|t|)^{\varepsilon\!}\!\lt\!M_1\!\lt\! \min\{ (M|t|)^{2/5\!},$ $(M|t|)^{1/2-8\delta}\}(M|t|)^{-\varepsilon}$ for any $\varepsilon\!>\!0$, where $M\!=\!M_1M_2$, $|t|\!\geq\! 1$, and $0\lt\delta\lt 1/52$. Then we have $$ L\left(\frac{1}{2}+it,\pi\otimes \chi\right)\ll_{\pi,\varepsilon} (M|t|)^{3/4-\delta+\varepsilon}. $$