By numerically solving the single-particle stationary Schrödinger equation and the Gross-Pitaevskii equation with mean-field interactions at zero temperature, we have investigated the ground state properties of the rotating spinorbital-angular-momentum coupled Bose-Einstein condensates in a harmonic trapping potential. The results show that the rotation lifts the double degeneracy of the single-particle energy spectrum in the angular momentum space, and leads to the vortex state with a single angular momentum. The angular momentum of the vortex depends on the rotating frequency, the strength of the laser beam, and the spin-orbital-angular-momentum coupling. In particular, if the rotating frequency is below a critical value, the angular momentum of the ground state vortex remains unaffected by the rotating frequency. While the rotating frequency surpasses the critical value, the angular momentum of the ground state vortex will increase with the rotating frequency. By assuming that the system is confined in a ring trap, the expression of the single-particle energy spectrum in the angular momentum space can be obtained, which clarifies how the rotation frequency affects the angular momentum of the ground state. In the presence of the atomic interactions, similar phenomena can also be observed in the mean-field ground state at zero temperature.