The normalized N0 (N0*) has been proposed as a key parameter to characterize the raindrop size distribution (DSD). The N0*, originally defined as a quantity proportional to the ratio of LWC divided by Dm^4, has interesting properties to reduce scatters among various integral-rain parameters and to discriminate the amplitude and shape of DSDs. The concept of N0* can be generalized by defining a mean drop diameter weighted by x-th moment of DSD, where x can be arbitrarily chosen. We found that x=4.2 is optimum to minimize the scatter between normalized Z and R both theoretically and experimentally. The optimum value of x depends on the combination of integral rain parameters of interest. This generalization can enhance the ability of N0* to consider an optimum set of DSD parameters for a specific problem of radar rainfall remote sensing, and the physical and practical meaning of the parameterization of DSD.