Quantum walks on oriented graphs exhibit different behavior than on undirected graphs. For example, perfect state transfer from $a$ to $b$ at time $t$ does not imply perfect state transfer from $b$ to $a$ at the same time, nor does it imply periodicity at vertex $a$ at time $2t$. But perfect state transfer from $a$ to $b$, or local uniform mixing at vertex $a$ does imply that the oriented graph is periodic at vertex $a$. It is known that an oriented graph $X$ is periodic at vertex $a$ if and only if all the eigenvalues of $A(X)$ (skew symmetric) in the support of $a$ are integer multiples of $\sqrt{\Delta}$ for some square-free integer $\Delta<0$. Here we give a characterization for when an oriented Cayley graph has such eigenvalues.