NON-VANISHING OF HECKE L-FUNCTIONS OF CUSP FORMS OF INTEGER AND HALF-INTEGER WEIGHT

Abstract

We show a non-vanishing result for L-functions of cuspidal Hecke eigenforms of integer weight in the full modular group and of half integer weight in the plus space. In chapter 1, we review definitions of modular forms of integer weight, their Hecke operators and their corresponding L-functions. In chapter 2, we introduce modular forms of half integer weight and some related properties. In chapter 3, will show that the average of the normalized L-functions L∗(f,s) with f a cusp form of weight k in SL2(Z), running over a basis of Hecke eigenforms, does not vanish inside the critical strip. A similar result will be presented in chapter 4 for cusp forms of half-integer weight in the plus space.