Although connectionist temporal classification (CTC) has the label context independence assumption, it can still implicitly learn a context-dependent internal language model (ILM) due to modern powerful encoders. In this work, we investigate the implicit context dependency modeled in the ILM of CTC. To this end, we propose novel context-dependent ILM estimation methods for CTC based on knowledge distillation (KD) with theoretical justifications. Furthermore, we introduce two regularization methods for KD. We conduct experiments on Librispeech and TED-LIUM Release 2 datasets for in-domain and cross-domain evaluation, respectively. Experimental results show that context-dependent ILMs outperform the context-independent priors in cross-domain evaluation, indicating that CTC learns a context-dependent ILM. The proposed label-level KD with smoothing method surpasses other ILM estimation approaches, with more than 13% relative improvement in word error rate compared to shallow fusion.