This paper conducted a novel study to explore the capabilities of ChatGPT, a state-of-the-art LLM, in static analysis tasks such as static bug detection and false positive warning removal. In our evaluation, we focused on two types of typical and critical bugs targeted by static bug detection, i.e., Null Dereference and Resource Leak, as our subjects. We employ Infer, a well-established static analyzer, to aid the gathering of these two bug types from 10 open-source projects. Consequently, our experiment dataset contains 222 instances of Null Dereference bugs and 46 instances of Resource Leak bugs. Our study demonstrates that ChatGPT can achieve remarkable performance in the mentioned static analysis tasks, including bug detection and false-positive warning removal. In static bug detection, ChatGPT achieves accuracy and precision values of up to 68.37% and 63.76% for detecting Null Dereference bugs and 76.95% and 82.73% for detecting Resource Leak bugs, improving the precision of the current leading bug detector, Infer by 12.86% and 43.13% respectively. For removing false-positive warnings, ChatGPT can reach a precision of up to 93.88% for Null Dereference bugs and 63.33% for Resource Leak bugs, surpassing existing state-of-the-art false-positive warning removal tools.