It is known that for a soluble extension of algebraic number fields $L:K$, the ratio of the Dedekind $\zeta$-functions $\frac{\zeta_L(s)}{\zeta_K(s)}$ is an entire function.
It is also conjectured (Artin) that the above holds for any Galois extension $L:K$.
My question is whether or not there exists some corresponding conjecture that applies for all (not necessarily Galois) field extensions.
$\textit{E.g.}$ if we let $K=\mathbb{Q}$ and $L=\mathbb{Q}(\sqrt[3]{2})$, so that $L:K$ is not a normal extension, can anything be said about the holomorphicity of $\frac{\zeta_L(s)}{\zeta_K(s)}$, and is it even meaningful to consider these extensions? If so, how come I have never seen the case of non-Galois extensions addressed in the literature?