  A mixed graph is said to be \textit{integral} if all the eigenvalues of its
Hermitian adjacency matrix are integer. The \textit{mixed circulant graph}
$Circ(\mathbb{Z}_n,\mathcal{C})$ is a mixed graph on the vertex set
$\mathbb{Z}_n$ and edge set $\{ (a,b): b-a\in \mathcal{C} \}$, where $0\not\in
\mathcal{C}$. If $\mathcal{C}$ is closed under inverse, then
$Circ(\mathbb{Z}_n,\mathcal{C})$ is called a \textit{circulant graph}. We
express the eigenvalues of $Circ(\mathbb{Z}_n,\mathcal{C})$ in terms of
primitive $n$-th roots of unity, and find a sufficient condition for
integrality of the eigenvalues of $Circ(\mathbb{Z}_n,\mathcal{C})$. For
$n\equiv 0 \Mod 4$, we factorize the cyclotomic polynomial into two irreducible
factors over $\mathbb{Q}(i)$. Using this factorization, we characterize
integral mixed circulant graphs in terms of its symbol set. We also express the
integer eigenvalues of an integral oriented circulant graph in terms of a
Ramanujan type sum, and discuss some of their properties.
