I have been studying the proof of Chebotarev's Density Theorem from the book "Problems in Algebraic number Theory" by Ram Murty and Esmonde and encountered a problem.
Let $n(H, \psi)$ be the order of the pole of $L(s, \psi; K/K^H)$ at $s=s_0$, where $s_0$ is a fixed complex number, $K/k$ is a Galois extension of algebraic number field and $K^H$ is the subfield of $K$ fixed by subgroup $H$ of $Gal(K/k)$. $\psi$ is a one dimensional character of $H$. $L(s, \psi; K/K^H)$ denotes the Artin L-function.
Also, a simplified conclusion of Artin's Reciprocity Law has been stated in the book as follows: for a nilpotent subgroup $H$ and a one-dimensional character $\psi$ of $H$, $L(s,\psi, K/K^H)$ coincides with $L(s, \chi)$ with $\chi$ a Hecke character of $K^H$.
In proving a crucial statement, the authors use the fact "if $H$ is abelian, $n(H, \psi) \geq 0$ by Artin's Reciprocity Law".
I am having a hard time in deducing this particular statement. As much as I have tried to understand, if we need to prove $n(H, \psi) \geq 0$ and since $H$ is abelian (hence, nilpotent), we have to prove that the order of the pole of the Hecke L-function $L(s, \chi)$ to be $\geq 0$. Does this imply we have to show that $L(s, \chi)$ does not have any zero for $Re(s) \geq 1$? If so, what is the idea to realize this statement?
Since I have no experience in class field theory, so I am looking for a basic explanation using the version of Artin Reciprocity law as described. Also, the statement is in the solution of exercise 11.4.7 of the book in case anyone wants to have a look.