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Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possiblywith elliptic fixed points, and let $\chi$ denote a finite dimensional unitaryrepresentation of the fundamental group of $M$. Let $\Delta$ denote thehyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$associated to $\chi$. From the spectral theory of $\Delta$, there are threedistinct sequences of numbers: The first coming from the eigenvalues of $L^{2}$eigenfunctions, the second coming from resonances associated to the continuousspectrum, and the third being the set of negative integers. Using thesesequences of spectral data, we employ the super-zeta approach to regularizationand introduce two super-zeta functions, $\Z_-(s,z)$ and $\Z_+(s,z)$ that encodethe spectrum of $\Delta$ in such a way that they can be used to define theregularized determinant of $\Delta-z(1-z)I$. The resulting formula for theregularized determinant of $\Delta-z(1-z)I$ in terms of the Selberg zetafunction, see Theorem 5.3, encodes the symmetry $z\leftrightarrow 1-z$, whichcould not be seen in previous works, due to a different definition of theregularized determinant.

Super-zeta functions and regularized determinants associated to cofinite Fuchsian groups with finite-dimensional unitary representations