Given the fact of a case, Legal Judgment Prediction (LJP) involves a series of sub-tasks such as predicting violated law articles, charges and term of penalty. We propose leveraging a unified text-to-text Transformer for LJP, where the dependencies among sub-tasks can be naturally established within the auto-regressive decoder. Compared with previous works, it has three advantages: (1) it fits in the pretraining pattern of masked language models, and thereby can benefit from the semantic prompts of each sub-task rather than treating them as atomic labels, (2) it utilizes a single unified architecture, enabling full parameter sharing across all sub-tasks, and (3) it can incorporate both classification and generative sub-tasks. We show that this unified transformer, albeit pretrained on general-domain text, outperforms pretrained models tailored specifically for the legal domain. Through an extensive set of experiments, we find that the best order to capture dependencies is different from human intuitions, and the most reasonable logical order for humans can be sub-optimal for the model. We further include two more auxiliary tasks: court view generation and article content prediction, showing they can not only improve the prediction accuracy, but also provide interpretable explanations for model outputs even when an error is made. With the best configuration, our model outperforms both previous SOTA and a single-tasked version of the unified transformer by a large margin. Code and dataset are available at https://github.com/oli-yun/Dependency-LJP.