Certain L-functions at s = 1/2

Introduction. The vanishing orders of L-functions at the centers of their functional equations are interesting objects to study as one sees, for example, from the Birch-Swinnerton-Dyer conjecture on the Hasse-Weil L-functions associated with elliptic curves over number fields. In this paper we study the central zeros of the following types of L-functions: (i) the derivatives of the Mellin transforms of Hecke eigenforms for SL₂(ℤ), (ii) the Rankin-Selberg convolution for a pair of Hecke eigenforms for SL₂(ℤ), (iii) the Dedekind zeta functions. The paper is organized as follows. In Section 1, the Mellin transform L(s,f) of a holomorphic Hecke eigenform f for SL₂(ℤ) is studied. We note that every L-function in this paper is normalized so that it has a functional equation under the substitution s ↦ 1-s. In Section 2, we study some nonvanishing property of the Rankin-Selberg convolutions at s=1/2. Section 3 contains Kurokawa's result asserting the existence of number fields such that the vanishing order of the Dedekind zeta function at s=1/2 goes to infinity.