# Values of Artin L-functions at negative integers

Let $F$ be a number field and $\chi$ a one dimensional Artin character. That is, it is a map $\chi: Gal(\overline F/F) \to \mathbb C^\times$ and let $L(s,\chi)$ be it's L-series.

What is known about the values of $L(s,\chi)$ at negative integers? Are they always algebraic integers contained in $F(\chi)$? When are they zero? Is there a (conjectured) explicit representation?

What about the case when $\chi$ is trivial? What about higher dimensional characters? Are there any conjectures dealing with these values?

I only know the case $F = \mathbb Q$ and $\chi$ a Dirichlet character where you can find an explicit representation in terms of (generalized) Bernoulli numbers.

The question of order of vanishing is quite elementary: the L-function of a Hecke character (i.e. 1-dimensional Artin representation) over any number field has an Euler product, which is convergent and non-vanishing for $s > 1$, and it satisfies a functional equation relating $L(\chi, s)$ to $L(\bar\chi, 1-s)$; hence the orders of vanishing for negative integer $s$ are completely determined by the Gamma-factors in the functional equation. After some unravelling, one finds that for $s < 0$, or for $s = 0$ and $\chi$ non-trivial, the order of vanishing is $$(\text{# of complex places of F}) + \left(\begin{array}{l}\text{# of real places of F}\\\text{where \chi has sign (-1)^s}\end{array}\right).$$
So there will only be non-vanishing values to study if $F$ is totally real, $\chi$ is either totally even (sign $+1$ at every real place) or totally odd (sign $-1$ at every real place), and $s$ satisfies a parity condition depending on the sign of $\chi$. These cases have been studied in great detail by many mathematicians, notably C.L. Siegel. It's known, for instance, that (for $s$ of the appropriate parity) $L(\chi, s)$ is an algebraic number and depends Galois-equivariantly on $\chi$; that is, we have $L(\chi, s)^\sigma = L(\chi^\sigma, s)$ for $\sigma \in \operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$. It is not always integral, but its denominator is very well understood (analogously to the von Staudt-Clausen theorem on Bernoulli numbers). Moreover, the crowning theorem of the subject, the Mazur--Wiles theorem (formerly Iwasawa's "main conjecture"), relates the numerator of $L(\chi, s)$ to the order of an ideal class group.
• "It is even worse for a field like Z[i] where there seem to be no critical values despite the values at negative integers being known explicitly". Yes, they're explicitly known to be zero! (There are interpretations of the leading terms at these points in terms of regulator maps on higher K-groups, but now we are in very deep water indeed. If this interests you, read Beilinson's 1984 article "Higher regulators and values of $L$-functions".) – David Loeffler Oct 19 '17 at 14:52