Can algebraic number theory be used to study L-rigs?

This question is a follow-up to Are there infinitely many L-rigs?, where I conjecture that every L-rig $mathcal{L}$ gives rise to an “L-field” $mathcal {K}_{mathcal{L}}$. Considering such an L-field as an extension of $mathcal{K}_{mathcal{L}_{0}}cong mathbb{Q}$, it should be possible to use the tools of algebraic number theory to study L-rigs.

In particular, can one study the decomposition of L-functions $F$ that are irreducible for the law $otimes$, and thus of prime degree $d_{F}$, into “L-ideals” of some “L-integers ring” $mathcal{O}_{mathcal{K}_{mathcal{L}}}$, this decomposition mimicking the decomposition of $d_{F}$ as an element of $mathcal{O}_{mathbb{Q}}$ into ideals of $mathcal{O}_{mathbb{K}}$ for some extension $mathbb{K}$ of $mathbb{Q}$?