Vol. 5, No. 8, 2011

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
The behavior of Hecke $L$-functions of real quadratic fields at $s=0$

Byungheup Jun and Jungyun Lee

Vol. 5 (2011), No. 8, 1001–1026
Abstract

For a family of real quadratic fields {Kn = (f(n))}n, a Dirichlet character χ modulo q, and prescribed ideals {bn Kn}, we investigate the linear behavior of the special value of the partial Hecke L-function LKn(s,χn := χ NKn,bn) at s = 0. We show that for n = qk + r, LKn(0,χn,bn) can be written as

1 12q2(Aχ(r) + kBχ(r)),

where Aχ(r),Bχ(r) [χ(1),χ(2),,χ(q)] if a certain condition on bn in terms of its continued fraction is satisfied. Furthermore, we write Aχ(r) and Bχ(r) explicitly using values of the Bernoulli polynomials. We describe how the linearity is used in solving the class number one problem for some families and recover the proofs in some cases.

Keywords
special values, Hecke L-functions, real quadratic fields, continued fractions
Mathematical Subject Classification 2000
Primary: 11M06
Milestones
Received: 7 March 2010
Revised: 24 March 2011
Accepted: 8 May 2011
Published: 5 June 2012
Authors
Byungheup Jun
School of Mathematics
Korea Institute for Advanced Study
Hoegiro 87, Dongdaemun-gu
Seoul 130-722
South Korea
Jungyun Lee
School of Mathematics
Korea Institute for Advanced Study
Hoegiro 87, Dongdaemun-gu
Seoul 130-722
South Korea