  We present the structure theorem for the positive support of the cube of the
Grover transition matrix of the discrete-time quantum walk (the Grover walk) on
a general graph $G$ under same condition. Thus, we introduce a zeta function on
the positive support of the cube of the Grover transition matrix of $G$, and
present its Euler product and its determinant expression. As a corollary, we
give the characteristic polynomial for the positive support of the cube of the
Grover transition matrix of a regular graph, and so obtain its spectra.
Finally, we present the poles and the radius of the convergence of this zeta
function.
