  We compute Benois $\mathscr{L}$-invariants of weight $1$ cuspforms and of
their adjoint representations and show how this extends Gross' $p$-adic
regulator to Artin motives which are not critical in the sense of Deligne.
Benois' construction depends on the choice of a regular submodule which is well
understood when the representation is $p$-regular, as it then amounts to the
choice of a ``motivic'' $p$-refinement. The situation is dramatically different
in the $p$-irregular case, where the regular submodules are parametrized by a
flag variety and thus depend on continuous parameters. We are nevertheless able
to show in some examples, how Hida theory and the geometry of the eigencurve
can be used to detect a finite number of choices of arithmetic and
``mixed-motivic'' significance.
