Zero-density estimates and fractional moments of Hecke L-functions

In this paper we study a relation between zero-density estimates of a Hecke L-function and fractional moments of the L  -function on or near Res=1/2. As a consequence, we prove that a bound of the type∫T2T|L(1/2+it)|2kdt≪T(logT)A for some 00A>0 is sufficient to deduce a universality theorem for Hecke L-functions.