  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.
