Integral moment of the Riemann zeta function and Hecke $L$ functions

Abstract

Let $f$ be a Hecke cusp form for $SL(2,\mathbb{Z})$. We prove an asymptotic formula for the mixed moment of the product of $\zeta (s)$ and $L(s,f)$ on the critical line. Similarly, we prove an asymptotic formula for the mixed moment of the product of $|{\zeta }^2(s)|$ and $L(s,f)$ on the critical line.