Power moments and zero density estimates of automorphic <i>L</i>-function

Let f be a normalized Maass cusp form for [Formula: see text]. For [Formula: see text], we define [Formula: see text] as the infimum of all numbers M [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the automorphic L-function attached to f. In this paper, we shall establish the upper bounds of [Formula: see text] and obtain the zero density estimates for [Formula: see text] as its applications.