Let $L/K$ be an abelian extension of number fields with Galois group $G$ and let $\chi : G \to \{\pm 1\}$ denote a real linear character of $G$. Denote $L(\chi,s)$ the Artin L-function associated to $\chi$, $n=\mathrm{ord}\_{s=0} L(\chi,s)$ and
$$
L^\ast(\chi,0)=\lim\_{s\to 0} L(\chi,s)s^{-n}
$$
its special value at $s=0$. It is known this L-functions is also the L-function associated to a Hecke character. My question is: what are the conjectures and results regarding the sign of $L^\ast(\chi,0)$ ? My current work has gotten me to conjecture that it is $-1$ for the trivial character and $1$ otherwise, and I wonder if this is something known or conjectured.



Sign of the special value at s=0 of Hecke L-functions
If $\chi$ is real-valued, then the question makes sense. Using the functional equation, it reduces to computing the sign of the non-zero real number $L(\chi, 1)$ if $\chi$ is non-trivial, or the residue of $L(\chi, s) = \zeta\_K(s)$ at $s = 1$ if $\chi$ is trivial. In either case, $L(\chi, s)$ tends to $+1$ for $s$ large and real, and it cannot vanish on $Re(s) > 1$, so $L(\chi, 1+\epsilon) > 0$ for all positive $\epsilon$. This shows that $L(\chi, 1) > 0$ for $\chi \ne 1$, and that $Res\_{s=1} \zeta\_K(s) < 0$.


