  We initiate the study of Selberg zeta functions $Z_{\Gamma,\chi}$ for
geometrically finite Fuchsian groups $\Gamma$ and finite-dimensional
representations $\chi$ with non-expanding cusp monodromy. We show that for all
choices of $(\Gamma,\chi)$, the Selberg zeta function $Z_{\Gamma,\chi}$
converges on some half-plane in $\mathbb{C}$. In addition, under the assumption
that $\Gamma$ admits a strict transfer operator approach, we show that
$Z_{\Gamma,\chi}$ extends meromorphically to all of $\mathbb{C}$.
