Investigation into the structural, elastic and electronic properties for pure zirconium (Zr) crystal had been conducted by the first-principles pseudopotential method based on density functional theory. Both methods, local density approximation (LDA) and generalized gradient approximation (GGA), had been applied on the geometrical optimization of pure Zr to address the difference between two methods and their applicabilities. The result elucidated LDA could match the experimental data better, compared with method GGA. What's more, the structural properties under pressure had been stimulated and analyzed, showing crystal lattice parameters and crystalline volume change nonlinearly within the external pressure. In contrast, the single point energy of Zr showed a great linear correlation with the changing pressure. The elastic constants of the pure Zr were calculated, proving that Zr would acquire excellent ductibility and mechanical stability under pressure. In addition, the optical properties of zirconium under different pressures were analyzed. The adsorbing coefficient increased with the increasing pressure.