We provide here a quantum mechanical investigation of the resonance states found in a study of conically intersecting electronic surfaces. The dynamical system under investigation consists of a bound electronic state having a conical intersection with a dissociative electronic state. Quantum mechanical resonances arise from the predissociation of vibrational states of the bound potential surface via the nonadiabatic coupling to the dissociative potential surface. Resonance energies and wave functions are computed using the complex coordinate method, and the resonances are characterized in terms of contributions from states of the uncoupled potential surfaces. Key results found in this study include the following: (i) there is no correlation between resonance positions and widths in that when the resonances are ordered by their positions, the corresponding widths (and lifetimes) fluctuate irregularly; (ii) the resonance energetically below the conical intersection cannot be identified as a tunneling resonance of the lowest adiabatic potential surface since its resonance lifetime is orders of magnitude larger than the tunneling lifetime; (iii) the resonance states (even those whose positions are energetically much higher than the conical intersection) are found to arise from a small number of vibrational states of the bound diabat coupling to each other via the continuum of the dissociative diabat; and (iv) none of the resonance states emanate from a bound state of the upper adiabatic cone-shaped potential surface. We also briefly investigate the resonance energies as a function of the nonadiabatic coupling strength; the irregular behavior of the resonance lifetimes with the coupling strength is a fingerprint of the conical intersection. Furthermore, we have performed a symmetry analysis of the resonances and introduced an effective Hamiltonian which, with the aid of a simple model, yields results in agreement with numerically exact results.