  It is well-known that the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac
operator on a closed Riemannian $\text{spin}^{\mathbb{C}}$ manifold $M^{2k}$ of
dimension $2k$ for $k \in \mathbb{N}$ is symmetric. In this article, we prove
that over an odd-dimensional Riemannian product $M_{1}^{2p} \times
M_{2}^{2q+1}$ with a product $\text{spin}^{\mathbb{C}}$ structure for $p \geq
1, q \geq 0$, the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator given
by a product connection is symmetric if and only if either the
$\text{spin}^{\mathbb{C}}$ Dirac spectrum of $M_{2}^{2q+1}$ is symmetric or
$(e^{\frac{1}{2}c_{1}(L_{1})} \hat{A}(M_1))[M_{1}]=0$, where $L_1$ is the
associated line bundle for the given $\text{spin}^{\mathbb{C}}$ structure of
$M_1$.
