Is this Siegel's formula correct?

Question

In the paper Zum Beweise des Starkschen Satzes Siegel considers the function

$$L_q(s)=\sum_{n=1}^{\infty}\left(\frac{q}{n}\right)n^{-s},$$

where $q$ is a discriminant of a quadratic number field and the character is the Kronecker symbol. Then he writes that "according to Dirichlet" we have, in case $G>0$,

$$L_G(1)=2G^{-1/2}h_G\log \varepsilon_G,$$

where $h_G$ is the corresponding class number and $\varepsilon_G$ the fundamental unit.

However, according to the book Zetafunktionen und quadratische Körper the formula reads

$$h_G=\frac{G^{1/2}}{\log \varepsilon_G}L_G(1).$$

Which one is correct? Am I making some mistake?

Answer 1

Dirichlet proved his class number formula for quadratic forms; in particular he was working with class numbers $h^+$ in the strict sense, and his unit $\varepsilon$ was the fundamental solution of the Pell equation $t^2 - Du^2 = 1$, not the fundamental unit $\eta$ of the corresponding number field. The relation $$ \eta^{2h} = \varepsilon^{h^+} $$ encodes the two cases

• $\varepsilon = \eta$, $h^+ = 2h$
• $\varepsilon = \eta^2$, $h^+ = h$.