Higher moments of automorphic L -functions of GL(r) and its applications

Suppose that π is a unitary cuspidal automorphic representation of GL_r(𝔸_ℚ) and L(s,π) is the automorphic L -function related to π , For 1/2<σ<1 , let m(σ) ≥ 2 be the supremum of all numbers m such that ∫_1^T|L(s,π)|^m dt ≪_π,ε T^1+ε. We are interested in the lower bound of m(σ) for 1-1/r<σ<1 . Then as an application, we establish asymptotic formulas for the second, fourth and sixth power moments of L(s,π) .