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

I want to modify it by adding a character $\psi$: 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)

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)