We report on a timing and spectral analysis of the young, high magnetic field rotation-powered pulsar (RPP) B1509-58 using Chandra continuous-clocking mode observation. The pulsar's X-ray light curve can be fit by the two Gaussian components and the pulsed fraction shows moderate energy dependence over the Chandra band. The pulsed X-ray spectrum is well described by a power law with a photon index 1.16(4), which is harder than the values measured with RXTE PCA and NuSTAR. This result supports

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... e log-parabolic model for the broadband X-ray spectrum. With the unprecedented angular resolution of Chandra, we clearly identified off-pulse X-ray emission from the pulsar, and its spectrum is best fit by a power law plus blackbody model. The latter component has a temperature of ~0.14 keV with a bolometric luminosity comparable to the luminosities of other young and high magnetic field RPPs, and it lies between the temperature of magnetars and typical RPPs. In addition, we found that the nonthermal X-ray emission of PSR B1509-58 is significantly softer in the off-pulse phase than in the pulsed phase, with the photon index varying between 1.0 and 1.8 and anticorrelated with the flux. This is similar to the behavior of three other young pulsars. We interpreted it as different contributions of pair-creation processes at different altitudes from the neutron star surface according to the outer-gap model.