#### Vol. 5, No. 8, 2011

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors' Addresses Editors' Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Contacts Author Index To Appear 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 ${\left\{{K}_{n}=ℚ\left(\sqrt{f\left(n\right)\phantom{\rule{0.3em}{0ex}}}\phantom{\rule{0.3em}{0ex}}\right)\right\}}_{n\in ℕ}$, a Dirichlet character $\chi$ modulo $q$, and prescribed ideals $\left\{{\mathfrak{b}}_{n}\subset {K}_{n}\right\}$, we investigate the linear behavior of the special value of the partial Hecke $L$-function ${L}_{{K}_{n}}\left(s,{\chi }_{n}:=\chi \circ {N}_{{K}_{n}},{\mathfrak{b}}_{n}\right)$ at $s=0$. We show that for $n=qk+r$, ${L}_{{K}_{n}}\left(0,{\chi }_{n},{\mathfrak{b}}_{n}\right)$ can be written as

$\frac{1}{12{q}^{2}}\left({A}_{\chi }\left(r\right)+k{B}_{\chi }\left(r\right)\right),$

where ${A}_{\chi }\left(r\right),{B}_{\chi }\left(r\right)\in ℤ\left[\chi \left(1\right),\chi \left(2\right),\dots ,\chi \left(q\right)\right]$ if a certain condition on ${\mathfrak{b}}_{n}$ in terms of its continued fraction is satisfied. Furthermore, we write ${A}_{\chi }\left(r\right)$ and ${B}_{\chi }\left(r\right)$ 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
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