Let $K$ be a quadratic imaginary field, $\bf n$ an ideal in the ring of integers ${\cal O}_K$ and $\xi$ an algebraic Hecke character of type $(A_0)$ for the modulus $\bf n$. One knows (from Weil) that there exists a number field $E=E_\xi\supseteq K$ with the property that $\xi$ takes values in $E^\times$.

Let $p$ be a prime that splits in $K$. Consider the following condition: there exists an unramified place $v\mid p$ in $E$ with residue field $k_v={\Bbb F}_p$ such that $\xi$ takes values in the group of $v$-units in $E$.

The condition implies the existence of a $p$-adic avatar of $\xi$ with values in ${\Bbb Z}_p^\times$.

I would like to know:

1) to the best of your knowledge, has been this condition considered somewhere? does it have a "name"?

2) I'm tempted to say that $\xi$ is $p$-split if the condition is satisfied (and that $v$ splits $\xi$). Would this name conflict with other situations that I should be aware of?