The explicit dispersion equation for a one-dimensional periodic structure with alternative layers of left-handed material (LHM) and right-handed material (RHM) is given and analyzed. Some unusual phenomena such as spurious modes with complex frequencies, discrete modes and photon tunneling modes are observed in the band structure. The existence of spurious modes with complex frequencies is a common problem in the calculation of the band structure for such a photonic crystal. Discrete modes may exist regardless whether the optical length of the LHM layer cancels the optical length of the RHM layer or not. Physical explanation and significance are given for the discrete modes (with real values of wave number) and photon tunneling propagation modes (with imaginary wave numbers in a limited region).