  The primary goal of this article is to study $p$-adic Beilinson conjectures
in the presence of exceptional zeros for Artin motives over CM fields. In more
precise terms, we address a question raised by Hida and Tilouine on the order
of vanishing of Katz $p$-adic $L$-functions associated to CM fields, by means
of a leading term formula we prove in terms of Rubin-Stark elements. In the
particular case when the CM field in question is imaginary quadratic, our
leading term formula and its consequences we record herein are unconditional.
