A singular representation of the complex-valued extinction coefficient of metal nanoparticles is developed to characterize the resonant behavior of plasmonic systems exhibiting an arbitrary number of resonances. This complex coefficient is analytically represented in form of a meromorphic function of the pulsation containing a singular (resonant) and a regular part, and an original algorithm based on a numerical derivation is proposed to find all resonant parameters of each excited mode. This approach, applied to silver nanoparticles, allows a characterization of the resonance red-shift and broadening when increasing the particle size or the local refractive index, as well as a particular sphere radius presenting a minimal bandwidth corresponding to minimal losses in the system. The optical cross sections of individual modes present an optimal particle size that maximizes the absorption cross section and from which the scattering process becomes predominant with respect to the absorption. Optical efficiencies can also be optimized regarding the particle size, and their variations are correlated to those of the maximum near-field intensity. The hybrid modes in silver dimers are also analyzed, and the hot spot intensity resulting from the longitudinal mode excitation can also be maximized by optimizing the particle size and the local refractive index.