We use a nearly complete sample of gamma-ray bursts (GRBs) detected by the Swift satellite to study the correlations between the spectral peak energy Epeak of the prompt emission, the isotropic energetics Eiso and the isotropic luminosity Liso. This GRB sample is characterized by a high level of completeness in redshift (90 per cent). This allows us to probe in an unbiased way the issue related to the physical origin of these correlations against selection effects. We find that one burst, GRB 061021, is an outlier to the Epeak–Eiso correlation. Despite this case, we find strong Epeak–Eiso and Epeak–Liso correlations for the bursts of the complete sample. Their slopes, normalizations and dispersions are consistent with those found with the whole sample of bursts with measured redshift and Epeak. This means that the biases present in the total sample commonly used to study these correlations do not affect their properties. Finally, we also find no evolution with redshift of the Epeak–Eiso and Epeak–Liso correlations.