Artin $L$-functions and abelianization

Question

Let $L/K$ be a finite Galois extension of global fields, with $G=\mathrm{Gal}(L/K)$. Let $H \leq G$ and $\chi:H \to \mathbb{C}$ a non-trivial irreducible character of $H$. Then we can define the (Artin) $L$-function $$L(L^H,\chi,s)$$ in the usual way. Suppose that $N$ is a normal subgroup of $G$ containing $\ker(\chi)$. Then there is an isomorphism $$\varphi:HN/N \to H/(H \cap N).$$ We can pull back $\varphi^*\chi$ to obtain a non-trivial irreducible character of $HN/N$. There is then an associated $L$-function $$L(L^{HN},\varphi^*\chi,s).$$ My question: What can we say, if anything, about the relationship between these two $L$-functions? I am mainly interested in the case where $N=[G,G]$ and $\chi$ is a linear character of $H$ (i.e., if I replace $L^H$ with the corresponding subfield of $L^\mathrm{ab}$, how does the $L$-function of $\chi$ change?).