Certain L-functions at s = 1/2

Shin-ichiro Mizumoto

Acta Arithmetica (1999)

  • Volume: 88, Issue: 1, page 51-66
  • ISSN: 0065-1036

Abstract

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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.

How to cite

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Shin-ichiro Mizumoto. "Certain L-functions at s = 1/2." Acta Arithmetica 88.1 (1999): 51-66. <http://eudml.org/doc/207231>.

@article{Shin1999,
abstract = { 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. },
author = {Shin-ichiro Mizumoto},
journal = {Acta Arithmetica},
keywords = {central zero; integral representations; nonvanishing; modular -series; Rankin-Selberg convolutions; holomorphic projection operator},
language = {eng},
number = {1},
pages = {51-66},
title = {Certain L-functions at s = 1/2},
url = {http://eudml.org/doc/207231},
volume = {88},
year = {1999},
}

TY - JOUR
AU - Shin-ichiro Mizumoto
TI - Certain L-functions at s = 1/2
JO - Acta Arithmetica
PY - 1999
VL - 88
IS - 1
SP - 51
EP - 66
AB - 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.
LA - eng
KW - central zero; integral representations; nonvanishing; modular -series; Rankin-Selberg convolutions; holomorphic projection operator
UR - http://eudml.org/doc/207231
ER -

References

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