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

Abstract

For a family of real quadratic fields ${\left\{K_n=\mathbb{Q}\left(\sqrt{f\left(n\right)}\right)\right\}}_{n\in \mathbb{N}}$, 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

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

where $A_{\chi }\left(r\right),B_{\chi }\left(r\right)\in \mathbb{Z}\left[\chi \left(1\right),\chi \left(2\right),\dots ,\chi \left(q\right)\right]$ 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.