[en] We characterized a BNL 1.3GHz half-cell SRF gun is tested for GaAs photocathode. The gun already was simulated several years ago via two-dimensional (2D) numerical codes (i.e., Superfish and Parmela) with and without the beam. In this paper, we discuss our investigation of its characteristics using a three dimensional (3D) full-wave code (CST STUDIO SUITE(trademark)).The input/pickup couplers are sited symmetrically on the same side of the gun at an angle of 180^o. In particular, the inner conductor of the pickup coupler is considerably shorter than that of the input coupler. We evaluated the cross-talk between the beam (trajectory) and the signal on the input coupler compared our findings with published results based on analytical models. The CST STUDIO SUITE(trademark) also was used to predict the field within the cavity; particularly, a combination of transient/eigenmode solvers was employed to accurately construct the RF field for the particles, which also includes the effects of the couplers. Finally, we explored the beam's dynamics with a particle in cell (PIC) simulation, validated the results and compare them with 2D code result.