On the coefficients of $\ell$-fold product $L$-function

Abstract

Let $f \in S_{k}(SL_2(\mathbb{Z}))$ be a normalized Hecke eigenforms of integral weight $k$ for the full modular group. In the article, we study the average behaviour of Fourier coefficients of $\ell$-fold product $L$-function. More precisely, we establish the asymptotics of power moments associated to the sequence $\{\lambda_{f \otimes f \otimes \cdots \otimes_{\ell} f}(n)\}_{n- {\rm squarefree}}$ where ${f \otimes f \otimes \cdots \otimes_{\ell} f}$ denotes the $\ell$-fold product of $f$. As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences for odd $\ell$-fold product $L$-function. A similar result also holds for the sequence $\{\lambda_{f \otimes f \otimes \cdots \otimes_{\ell} f}(n)\}_{n \in \mathbb{N}}$.