Thinking from the automorphic point of view, it should be an $L$-function over $\mathbb Q$ corresponding to the automorphic induction of the Hecke character to a representation $\Pi$ of $GL_N(\mathbb A_{\mathbb Q})$ where $N=deg(E:\mathbb Q)$. However, automorphic induction is only known for cyclic (whence solvable) and non-normal cubic extensions.

Depending on your extension and what you induce, it may be that $\Pi$ is cuspidal, but in general it should break up as an isobaric sum
$$ \Pi = \pi_1 \boxplus \cdots \boxplus \pi_r,$$
where the $\pi_i$'s are cuspidal. ($r=1$ and $\pi_1 = \Pi$ if $\Pi$ is cuspidal). Then your desired $L$-function decomposition over $\mathbb Q$ is
$$ L(s,\pi_1)L(s,\pi_2) \cdots L(s,\pi_r).$$

In the case of the Dedekind zeta function for an abelian extension, these $\pi_i$'s correspond to the irreducible characters of Gal$(E/\mathbb Q)$. For more general extensions, one knows this when each irreducible representation of Gal$(E/\mathbb Q)$ is known to be modular (corresponding to a cuspidal automorphic representation).

3more comments