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)



Special values of Hecke L-function
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).


