This work reports on the plasmonic properties of an eccentric nanostructure consisting of a silver ring and a gold disk, in the frequency regime of several hundreds of TetraHertz. In the structure, a gold nanodisk is located in a silver nanoring, with the centers of the two metals being set apart. As the radius of the gold disk, the separation between the centers, the light polarization, and the refractive index of the surrounding dielectric material are respectively varied, the normalized scattering cross section of the system is numerically simulated, using the finite difference time domain method. Multiple plasmonic resonances are observed, the plasmonic coupling effect between the ring and the disk is revealed, and the corresponding electromagnetic field distributions are demonstrated. The resonance peak associated with the coupling is found to possess a Fano-like lineshape, and it is well fitted to a model that employs a Fano function. The resonance frequency, the spectral width, as well as the characteristic q factor are determined from the best fit parameters, and the plasmonic characteristics of the eccentric nanostructure are quantitatively probed. The results in this work may be of significance in designs of plasmonic devices that are based on bi-metallic eccentric structures.