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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.

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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.

For higher-dimensional Artin representations, much less is known (although there are plenty of very precise conjectures). We don't even know if the L-function has analytic continuation, only meromorphic continuation, so asking about its values at negative integers might not even make sense.

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    $\begingroup$ Thanks a lot. I thought we knew that any poles of the meromorphic continuation would only have to occur at 1, if at all. If so, the values at negative integers should still be meaningful, right? Also, this might be too much to ask for but what is known in the case of Hasse-Weil zeta functions (say, for an elliptic curve)? Do you have a reference for the conjectures in the general higher dimensional representations case (and maybe even the Hasse-Weil case)? $\endgroup$ – Asvin Oct 19 '17 at 10:52
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    $\begingroup$ Sorry, yes, you are right about the meromorphic continuation issue; the proof using Brauer's induction theorem does indeed give you that the meromorphic continuation has the "right" order of vanishing at integer points (and indeed everywhere outside the critical strip). For L-functions of more general arithmetic objects (motives), e.g. Hasse--Weil, you should read Deligne's article in the Corvallis proceedings. $\endgroup$ – David Loeffler Oct 19 '17 at 11:02
  • $\begingroup$ Is that article the one where he defines the notion of a critical value for L-functions? If I understand correctly, this requires that the archimedean factor of the motive and it's dual not have a pole at the point. However, even in the case of the Dedekind zeta function, this seems to ignore the negative even integers even though the L function vanishes at that point. 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. $\endgroup$ – Asvin Oct 19 '17 at 12:41
  • $\begingroup$ Does Deligne's article talk about values at non critical points or perhaps for an Elliptic curve, the only interesting points are the critical points? $\endgroup$ – Asvin Oct 19 '17 at 12:42
  • $\begingroup$ "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".) $\endgroup$ – David Loeffler Oct 19 '17 at 14:52

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