Large language models (LLMs) are increasingly integrated into data management systems, yet their energy consumption remains largely unexplored. EcoLLM is a workload-centric benchmark for studying the energy-aware trade-offs of LLMs in data-processing tasks, modeling workload families, operator complexity, and data scale, and supporting both local and API-based models through a hybrid energy measurement methodology. Our results show that energy efficiency and task effectiveness are not aligned, and reveal a consistent trade-off between latency and energy, where lower latency is often achieved at disproportionately higher energy cost.