Factorization of the Dedekind zeta function of an abelian number field

The Dedekind zeta function of an abelian number fieldfactors as a product of Dirichlet L-functions as follows. Let $K$be an abelian number field, i.e. $K/\mathbb{Q}$ is Galois and$\mathrm{Gal}(K/\mathbb{Q})$ is abelian. Then, by the Kronecker-Weber theorem,there is an integer $n$ (which we choose to be minimal) such that$K\subseteq \mathbb{Q}({\zeta }_n)$ where ${\zeta }_n$ is a primitive $n$throot of unity. Let $G=\mathrm{Gal}(\mathbb{Q}({\zeta }_n)/\mathbb{Q})\cong {(\mathbb{Z}/n\mathbb{Z})}^{\times }$ and let $\chi :G\to {ℂ}^{\times }$ be aDirichlet character. Then the kernel of $\chi$ determines a fixedfield of $\mathbb{Q}({\zeta }_n)$. Further, for any field $K$ as before,there exists a group $X$ of Dirichlet characters of $G$ such that$K$ is equal to the intersection of the fixed fields by thekernels of all $\chi \in X$. The order of $X$ is $[K:\mathbb{Q}]$ and$X\cong \mathrm{Gal}(K/\mathbb{Q})$.

Theorem ([1], Thm. 4.3).

Let $K$ be an abelian number field and let $X$ bethe associated group of Dirichlet characters. The Dedekind zetafunction of $K$ factors as follows:

$${\zeta }_K(s)=\prod _{\chi \in X}L(s,\chi ).$$

Notice that for the trivial character ${\chi }_0$ one has$L(s,{\chi }_0)=\zeta (s)$, the Riemann zeta function, which has asimple pole at $s=1$ with residue $1$. Thus, for an arbitraryabelian number field $K$:

$${\zeta }_K(s)=\prod _{\chi \in X}L(s,\chi )=\zeta (s)\cdot \prod _{{\chi }_0\ne \chi \in X}L(s,\chi )$$

where the last product is taken over allnon-trivial characters $\chi \in X$.