The Zeta Functions of Complexes from $\text{Sp}(4)$

Let $F$ be a non-archimedean local field with a finite residue field. To a 2-dimensional finite complex $X_{\mathrm{\Gamma }}$ arising as the quotient of the Bruhat-Tits building $X$ associated to ${\text{Sp}}_4(F)$ by a discrete torsion-free cocompact subgroup $\mathrm{\Gamma }$ of ${\text{PGSp}}_4(F)$, associate the zeta function $Z(X_{\mathrm{\Gamma }},u)$ which counts geodesic tailless cycles contained in the 1-skeleton of $X_{\mathrm{\Gamma }}$. Using a representation-theoretic approach, we obtain two closed form expressions for $Z(X_{\mathrm{\Gamma }},u)$ as a rational function in $u$. Equivalent statements for $X_{\mathrm{\Gamma }}$ being a Ramanujan complex are given in terms of vertex, edge, and chamber adjacency operators, respectively. The zeta functions of such Ramanujan complexes are distinguished by satisfying the Riemann Hypothesis.