Localized basis sets such as Gaussians[1], truncated pseudo-atomic orbitals[2,3], real-space grids[4,5,6,7,8,9], B-spline (or ``blip'') functions[10], and wavelets[11], have been used in first-principles calculations. Recently there has been a surge of activity to investigate linear-scaling methods[12] (where the computational effort and memory requirement scale linearly with the system size), all of which use localized basis sets in their implementations. One localized basis set that was introduced for linear-scaling methods, the spherical-wave basis set[13], is interesting because while sharing some of the properties (such as the concept of energy cutoff) with the extended plane-wave basis set, it possesses other advantages such as each basis function being fully localized within a sphere. Even though it has been used to implement a linear-scaling method and tested against bulk crystalline silicon[14], this basis set has not yet been fully investigated. This work serves to reveal the properties of this localized basis set using a matrix diagonalization approach, which frees us from having to consider other sources of error introduced by other cutoffs (such as the density-matrix spatial cutoff[15,16,17]). The completeness and appropriateness of this basis set are investigated in first-principles calculations within density-functional theory through applications to molecules and bulk crystalline silicon. The remainder of this work is organized as follows. Section 2 introduces the spherical-wave basis set. In section 3, a brief description is given of the first-principles calculation within density-functional theory using the spherical-wave basis set. Section 4 contains the results of calculations on different test systems, which are compared with those obtained using the same theory level and approximations, but with a plane-wave basis set[18]. Section 5 contains the summary and conclusions.