Issue solving aims to generate patches to fix reported issues in real-world code repositories according to issue descriptions. Issue localization forms the basis for accurate issue solving. Recently, LLM-based issue localization methods have demonstrated state-of-the-art performance. However, these methods either search from files mentioned in issue descriptions or in the whole repository and struggle to balance the breadth and depth of the search space to converge on the target efficiently. Moreover, they allow LLM to explore whole repositories freely, making it challenging to control the search direction to prevent the LLM from searching for incorrect targets. This paper introduces CoSIL, an LLM-driven, powerful function-level issue localization method without training or indexing. CoSIL employs a two-phase code graph search strategy. It first conducts broad exploration at the file level using dynamically constructed module call graphs, and then performs in-depth analysis at the function level by expanding the module call graph into a function call graph and executing iterative searches. To precisely control the search direction, CoSIL designs a pruner to filter unrelated directions and irrelevant contexts. To avoid incorrect interaction formats in long contexts, CoSIL introduces a reflection mechanism that uses additional independent queries in short contexts to enhance formatted abilities. Experiment results demonstrate that CoSIL achieves a Top-1 localization accuracy of 43.3\% and 44.6\% on SWE-bench Lite and SWE-bench Verified, respectively, with Qwen2.5-Coder-32B, average outperforming the state-of-the-art methods by 96.04\%. When CoSIL is integrated into an issue-solving method, Agentless, the issue resolution rate improves by 2.98\%--30.5\%.