The intramolecular relaxation dynamics of unconcatenated ring polymers in dilute solutions is theoretically investigated within the framework of the Rouse-Zimm theory. The excluded volume interactions (EVIs) between the nonbonded monomers are modeled by a harmonic potential, where the interaction parameter is evaluated from Flory's mean-field approach. The hydrodynamic interactions (HIs) between the pairs of monomers are approximated by a preaveraged Oseen tensor. The mechanical moduli are dominated by the smaller relaxation rates corresponding to the collective relaxation modes in the low frequency regime, while they are governed by the higher relaxation rates corresponding to the local relaxation modes in the high frequency regime. EVI decreases the relaxation rates of the normal modes where the decrease for the collective modes is larger than that for the local modes, which consequently expands the width of the relaxation spectrum. The characteristic overall relaxation time is evaluated from the inverse of the crossover frequency, which is the same for rings of various sizes with HI and with both HI and EVI, while it shifts to lower frequencies with increasing ring size for the Rouse rings.