Algebra & Number Theory

The behavior of Hecke $L$-functions of real quadratic fields at $s=0$

Byungheup Jun and Jungyun Lee

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

1 1 2 q 2 ( A χ ( r ) + k B χ ( 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.

Article information

Algebra Number Theory, Volume 5, Number 8 (2011), 1001-1026.

Received: 7 March 2010
Revised: 24 March 2011
Accepted: 8 May 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$

special values Hecke L-functions real quadratic fields continued fractions


Jun, Byungheup; Lee, Jungyun. The behavior of Hecke $L$-functions of real quadratic fields at $s=0$. Algebra Number Theory 5 (2011), no. 8, 1001--1026. doi:10.2140/ant.2011.5.1001.

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