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

Suppose that π is a unitary cuspidal automorphic representation of G L 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 d t ≪ 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 ) .