Special values of Hecke L-function

Question

The Dedekind zeta function for a number field $K$ is defined as $\zeta_K(s)=\sum_{I\subset O_K} (N_{K/\mathbb{Q}}(I))^{-s}$.

By attaching a Hecke character $\psi$, we can define $L(s,\psi)=\sum_{I\subset O_K} \psi(I)(N_{K/\mathbb{Q}}(I))^{-s}$.

Is there any way to calculate the special value of such $L(s,\psi)$ at $1$ when $\psi$ is nontrivial? Trying to do it numerically in Sage but do not have any idea where to start.

I am particularly interested in cases when $K$ is totally real, $\psi$ is a product of sign functions of the embeddings. Two examples:

1)$K=\mathbb{Q}(\sqrt{3})$ and $\psi(x)=sgn(Norm(x))$

2)$K=\mathbb{Q}(\alpha)$ the cubic extension given by $\alpha^3 + 2\alpha^2 - 3\alpha - 1=0$. $\alpha$ has 3 real embeddings, two negative ones called $\sigma_1$ and $\sigma_2$ and one positive $\sigma_3$. Now let $\psi(x)=sgn(\sigma_1(x)\sigma_2(x))$ (One can verify it is well-defined on the class group b/c it is 1 on the units)

Answer 1

I think the following Magma code gives you what you want:

> K:=QuadraticField(3);
> G:=HeckeCharacterGroup(1*Integers(K),[1,2]);
> psi:=G.1;
> L:=LSeries(psi);
> Evaluate(L,1);

Note that in fact $L(\psi)=L(\chi_{-3})\cdot L(\chi_{-4})$.

> K:=NumberField(Polynomial([-1,-3,2,1]));
> [Sign(Real(e)) : e in Conjugates(K.1)]; // [1,-1,-1]
> G:=HeckeCharacterGroup(1*Integers(K),[2,3]);
> psi:=G.1;
> L:=LSeries(psi);
> Evaluate(L,1);

The field is nonabelian here, but you still might be able to write $L(\psi)$ in terms of Artin representations (which are 1-dim, thus Dirichlet chars, in the previous case). Indeed, the Conductor of $L(\psi)$ is 257, and it is valuable to search the LMFDB for dimension 3 Artin representations of this conductor.

http://www.lmfdb.org/L/ArtinRepresentation/3/257/2/

The LMFDB searching mechanism lists two Artin representations of this dimension and conductor, but they have the same $L$-function (compare the computed zeros).