# Certain L-functions at s = 1/2

Acta Arithmetica (1999)

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

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topShin-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 -

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