By means of transfer matrix method, the photonic band-gap and the center tunneling mode of onedimensional Sinc function photonic crystal with dispersive defect are discussed. The dispersive defects are described by a Lorentz oscillator model. The complex effective index of the structure is calculated. It is found that an extremely wide photonic band gap is obtained after adding defect layer. After the introduction of the dispersion of the defect layer, the dispersive defect modes are appeared near the center angular frequency. With the increase of the oscillator strength, the angular frequency of the gain defect mode is stepped, the transmittance of them is significantly increased, but the transmittance of the absorptive defect mode is reduced, and the slope of a nearly vertical of dispersion curves is obtained at the angular frequency of the dispersive defect mode,group velocity here is greatly reduced. The angular frequency of the defect mode is significantly affected by the refractive index of the substrate of the gain defect layer. The angular frequency of the defect mode is the lowest and transmittance of them is the maximum when the defect layer is located in the center of the periodic structure. The angular frequency of the defect mode is made redshift with the increasing of the angle of incidence, and the transmittance of the gain defect mode is obtained thousands times in a special angle of incidence.