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

Abstract

Let $π$ be a Hecke-Maass cusp form for $SL(3,\mathbb{Z})$ and $χ=χ_1χ_2$ a Dirichlet character with $χ_i$ primitive modulo $M_i$. Suppose that $M_1$, $M_2$ are primes such that $\max\{(M|t|)^{1/3+2δ/3},M^{2/5}|t|^{-9/20}, M^{1/2+2δ}|t|^{-3/4+2δ}\}(M|t|)^{\varepsilon}<M_1< \min\{ (M|t|)^{2/5},(M|t|)^{1/2-8δ}\}(M|t|)^{-\varepsilon}$ for any $\varepsilon>0$, where $M=M_1M_2$, $|t|\geq 1$ and $0<δ< 1/52$. Then we have $$ L\left(\frac{1}{2}+it,π\otimes χ\right)\ll_{π,\varepsilon} (M|t|)^{3/4-δ+\varepsilon}. $$

26 pages. Comments are welcome!