  This contribution gives an extensive study on spectra of mixed graphs via its
Hermitian adjacency matrix of the second kind { ($N$-matrix for short)}
introduced by Mohar \cite{0001}. This matrix is indexed by the vertices of the
mixed graph, and the entry corresponding to an arc from $u$ to $v$ is equal to
the sixth root of unity $\omega=\frac{1+{\bf i}\sqrt{3}}{2}$ (and its symmetric
entry is $\bar{\omega}=\frac{1-{\bf i}\sqrt{3}}{2}$); the entry corresponding
to an undirected edge is equal to 1, and 0 otherwise. The main results of this
paper include the following: {equivalent} conditions for a mixed graph that
shares the same spectrum of its $N$-matrix with its underlying graph are given.
A sharp upper bound on the spectral radius is established and the corresponding
extremal mixed graphs are identified. Operations which are called two-way and
three-way switchings are discussed--they give rise to some cospectral mixed
graphs. We extract all the mixed graphs whose rank of its $N$-matrix is $2$
(resp. 3). Furthermore, we show that {if $M_G$ is a connected mixed graph with
rank $2,$ then $M_G$ is switching equivalent to each connected mixed graph to
which it is cospectral}. However, this does not hold for some connected mixed
graphs with rank $3$. We identify all mixed graphs whose eigenvalues of its
$N$-matrix lie in the range $(-\alpha,\, \alpha)$ for
$\alpha\in\left\{\sqrt{2},\,\sqrt{3},\,2\right\}$.
