Let we have $K= \mathbb{Q}(\sqrt d)$ . For a rational prime $p$, we have the following cases if we consider its ramification in $O_K$:
$p$ is ramified if $p| \Delta$.
$p$ splits if $(\cfrac{d}{p}) = 1$
$p$ is inert if $(\cfrac{d}{p}) = -1$
Now let us define a character modulo $\Delta$, we know that $\Delta$ is $d$ if $d \equiv 1 \mod 4$ and $4d$ if $d \equiv 3 \mod 4$ .
$\chi(p) = 0$ if $p$ is ramified.
$\chi(p)=1$ if $p$ splits if $(\cfrac{d}{p}) = 1$
$\chi(p) = -1$ if p is inert if $(\cfrac{d}{p}) = -1$
From now on I have some problems understanding $\chi(.)$, the rest is my thinking:
$\chi(p)$ is actually $(\cfrac{d}{p}) $. I am not sure what this symbol is. I think that it is not Legendre symbol. Probably it is Kronoecker symbol because otherwise $(\cfrac{d}{2})$ is meaningless.
For example when we have $\mathbb{Q}[\sqrt{2}]$ and $\mathbb{Q}[\sqrt{5}]$ what is the corresponding $L-$series $L(\chi,s)$ ?
P.s. I do not know how to prove that $\chi(.)$ is a character. To add, they are $8$ periodic and $5$ periodic since their discriminants are $8$ and $5$ respectively.