The electronic and magnetic states as well as the structural and elastic properties of the equiatomic UZr alloy are investigated by first-principles calculations based on the density-functional theory with generalized gradient approximation (+U) approach. The strong correlation effects of the 5f electrons are studied. Results show that the orthorhombic structure with ferromagnetic configuration is the most stable state. The elastic constants show that the orthorhombic phase is mechanically stable while the γ(U,Zr) phase is unstable under ambient condition.