Algebra & Number Theory
- Algebra Number Theory
- Volume 5, Number 8 (2011), 1001-1026.
The behavior of Hecke $L$-functions of real quadratic fields at $s=0$
For a family of real quadratic fields , a Dirichlet character modulo , and prescribed ideals , we investigate the linear behavior of the special value of the partial Hecke -function at . We show that for , can be written as
where if a certain condition on in terms of its continued fraction is satisfied. Furthermore, we write and 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.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
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. https://projecteuclid.org/euclid.ant/1513729728