Traffic light control (TLC) with transit signal priority (TSP) is an effective way to deal with urban congestion and travel delay. The growing amount of available connected vehicle data offers opportunities for signal control with transit priority, but the conventional control algorithms fall short in fully exploiting those datasets. This paper proposes a novel approach for dynamic TLC with TSP at an urban intersection. We propose a deep reinforcement learning based framework JenaRL to deal with the complex real-world intersections. The optimisation focuses on TSP while balancing the delay of all vehicles. A two-layer state space is defined to capture the real-time traffic information, i.e. vehicle position, type and incoming lane. The discrete action space includes the optimal phase and phase duration based on the real-time traffic situation. An intersection in the inner city of Jena is constructed in an open-source microscopic traffic simulator SUMO. A time-varying traffic demand of motorised individual traffic (MIT), the current TLC controller of the city, as well as the original timetables of the public transport (PT) are implemented in simulation to construct a realistic traffic environment. The results of the simulation with the proposed framework indicate a significant enhancement in the performance of traffic light controller by reducing the delay of all vehicles, and especially minimising the loss time of PT.