  We introduce a twisted version of the Kawazumi-Zhang invariant $a_g(C) =
\varphi(C)$ of a compact Riemann surface $C$ of genus $g \geq 1$, and discuss
how it is related to the first Mumford-Morita-Milller class $e_1 = \kappa_1$ on
the moduli space of compact Riemann surfaces and the original Kawazumi-Zhang
invariant.
