We collect and reanalyze about 200 GRB data of prompt-emission with known redshift observed until the end of 2009, and select 101 GRBs which were well observed to have good spectral parameters to determine the spectral peak energy ($E_p$), 1-second peak luminosity ($L_p$) and isotropic energy ($E_{rm iso}$). Using our newly-constructed database with 101 GRBs, we first revise the $E_p$--$L_p$ and $E_p$--$E_{rm iso}$ correlations. The correlation coefficients of the revised correlations are 0.889 for 99 degree of freedom for the $E_p$--$L_p$ correlation and 0.867 for 96 degree of freedom for the $E_p$--$E_{rm iso}$ correlation. These values correspond to the chance probability of $2.18 times 10^{-35}$ and $4.27 times 10^{-31}$, respectively. It is a very important issue whether these tight correlations are intrinsic property of GRBs or caused by some selection effect of observations. In this paper, we examine how the truncation of the detector sensitivity affects the correlations, and we conclude they are surely intrinsic properties of GRBs. Next we investigate origins of the dispersion of the correlations by studying their brightness and redshift dependence. Here the brightness (flux or fluence) dependence would be regarded as an estimator of the bias due to the detector threshold. We find a weak fluence-dependence in the $E_p$--$E_{rm iso}$ correlations and a redshift dependence in the $E_p$--$L_p$ correlation both with 2 $sigma$ statistical level. These two effects may contribute to the dispersion of the correlations which is larger than the statistical uncertainty. We discuss a possible reason of these dependence and give a future prospect to improve the correlations.
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