  Let $p$ and $q$ be two distinct odd primes. Let $K$ be an imaginary quadratic
field over which $p$ and $q$ are both split. Let $\Psi$ be a Hecke character
over $K$ of infinity type $(k,j)$ with $0\le-j< k$. Under certain technical
hypotheses, we show that for a Zariski dense set of finite-order characters
$\kappa$ over $K$ which factor through the $\mathbb{Z}_q^2$-extension of $K$,
the $p$-adic valuation of the algebraic part of the $L$-value
$L(\overline{\kappa\Psi},k+j)$ is a constant independent of $\kappa$. In
addition, when $j=0$ and certain technical hypothesis holds, this constant is
zero.
