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

Abstract

Let $\ddot{I}€$ be a Hecke-Maass cusp form for $SL(3,\mathbb{Z})$ and $\ddot{I}‡=\ddot{I}{‡}_1\ddot{I}{‡}_2$ a Dirichlet character with $\ddot{I}{‡}_i$ primitive modulo $M_i$. Suppose that $M_1$, $M_2$ are primes such that $\max \{(M|t|{)}^{1/3+2\hat{I}´/3},M^{2/5}|t{|}^{-9/20},M^{1/2+2\hat{I}´}|t{|}^{-3/4+2\hat{I}´}\}(M|t|{)}^{\epsilon }<M_1<\min \{(M|t|{)}^{2/5},(M|t|{)}^{1/2-8\hat{I}´}\}(M|t|{)}^{-\epsilon }$ for any $\epsilon >0$, where $M=M_1{M}_2$, $|t|\ge 1$ and $0<\hat{I}´<1/52$. Then we have $$L\left(\frac{1}{2}+it,\ddot{I}€\otimes \ddot{I}‡\right){\ll }_{\ddot{I}€,\epsilon }(M|t|{)}^{3/4-\hat{I}´+\epsilon }.$$

26 pages. Comments are welcome!