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.