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Autonomous driving systems rely on precise trajectory prediction for safe and efficient motion planning. Despite considerable efforts to enhance prediction accuracy, inherent uncertainties persist due to data noise and incomplete observations. Many strategies entail formalizing prediction outcomes into distributions and utilizing variance to represent uncertainty. However, our experimental investigation reveals that existing trajectory prediction models yield unreliable uncertainty estimates, necessitating additional customized calibration processes. On the other hand, directly applying current calibration techniques to prediction outputs may yield sub-optimal results due to using a universal scaler for all predictions and neglecting informative data cues. In this paper, we propose Customized Calibration Temperature with Regularizer (CCTR), a generic framework that calibrates the output distribution. Specifically, CCTR 1) employs a calibration-based regularizer to align output variance with the discrepancy between prediction and ground truth and 2) generates a tailor-made temperature scaler for each prediction using a post-processing network guided by context and historical information. Extensive evaluation involving multiple prediction and planning methods demonstrates the superiority of CCTR over existing calibration algorithms and uncertainty-aware methods, with significant improvements of 11%-22% in calibration quality and 17%-46% in motion planning.