The Deligne–Ribet integrality conjecture for CM characters

Let $E$ be a totally real number field, and let $\chi:\mathbb A_E^\times/E^\times\rightarrow\bar{\mathbb Z}_p^\times$ be a finite-order character cutting out a non-real CM extension. Deligne–Ribet integrality conjecture. Then

$$L_E(s,\chi)\notin\bar{\mathbb Z}_p$$

if and only if both of the following hold: $\chi$ modulo $p$ is $\omega^{-1}$, and $\chi$ is associated with a character of $\operatorname{Gal}(E(\zeta_q)/E)$ for some power $q$ of $p$. This proposes a precise criterion for the exceptional non-integrality in the Deligne–Ribet theorem; it is open in the source.