On values of logarithmic derivative of L-function attached to modular form

Abstract

This paper aims to initiate a systematic investigation of the distribution of [Formula: see text], where [Formula: see text] is the [Formula: see text]-function attached to a normalized Hecke eigenform [Formula: see text] of weight [Formula: see text] for [Formula: see text] and [Formula: see text] is a primitive character. Assuming the Riemann hypothesis for [Formula: see text] and [Formula: see text], we prove an upper bound of [Formula: see text] in terms of [Formula: see text] and [Formula: see text]. This result is analogous to that of Ihara, Kumar Murty and Shimura. Next we show that the upper bound is not far from being optimal by proving an unconditional omega result which is analogous to a result of Mourtada and Kumar Murty. In course of proving the omega result, we prove a zero-density estimate for the [Formula: see text]-functions involved.