Let $K$ be an imaginary quadratic field and $D_K$ its discriminant. Further let $\mathcal O$ be an order in $K$ with conductor $f$ and

$$L(\chi,s)=\sum_{\mathfrak a}\chi(\mathfrak a)N(\mathfrak a)^{-s}$$

where the sum is over all integral ideals of $\mathcal O$ relatively prime to $f$, and $\chi$ is a character of the ideal group of $\mathcal O$ (ideals taken prime to $f$).

In his paper Zum Beweise des Starkschen Satzes, Siegel deduces the formula

$$\frac {1}{2\pi}f |D_K|^{1/2}L(\chi, 1)=-\sum_{\mathfrak C}\bar\chi(\mathfrak c)\log \left( \sqrt{\Im(\omega_{\mathfrak c})\eta(\omega_{\mathfrak c})^2}\right).$$ Here the sum on the right is over ideal classes in $\mathcal O$ (again prime to $f$), $\eta$ is the dedekind eta function, and $\omega_\mathfrak c=-\beta/\alpha$ if $\mathfrak c= [\alpha,\beta]$. This should follow from the Kronecker's first limit formula. Is there some English reference for the Siegel's formula, and the mentioned $L$-functions for non-maximal orders?

  • A representative of $\mathfrak{c}$ is a lattice $ u \mathbb{Z}+v \mathbb{Z}, u/v = a+ib$ and you can look at $h(x) = \sum_{n \in \mathbb{Z}^2} \exp(-\pi x \|\begin{pmatrix} 1 & a \\ 0 & b \end{pmatrix} n\|^2)$. The Poisson summation formula tells you $h(1/x) = |b|^{-1/2} x^{-1} \sum_{n \in \mathbb{Z}^2} \exp(-\pi x \| \begin{pmatrix} 1 & a \\ 0 & b \end{pmatrix}^{-\top} n\|^2)$ (which is related to the sum for a representative of $\mathfrak{c}^{-1}$). From this you know the asymptotic of $h$ as $x \to 0$ and the pole of its Dirichlet series, that you can relate to $\log \eta(u/v)$ – reuns Nov 9 at 0:11
  • @reuns Could you please elaborate on your comment? – Shimrod Nov 9 at 16:58

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