A twisted invariant of a compact Riemann surface

We introduce a twisted version of the Kawazumi-Zhang invariant $a_g(C)=\phi (C)$ of a compact Riemann surface $C$ of genus $g\ge 1$, 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.