We propose a new kind of partially coherent beams whose non-conventional correlation function, also called spectral degree of coherence (SDOC), contains two nonconventional components, i.e., a cosine and a Lorentz functions. Such beam meets the sufficient condition established by Gori, and thus it is physically realizable. Analytical expressions of the cross-spectral density function of the proposed beam passing through a paraxial ABCD optical system are derived based on the generalized Huygens-Fresnel diffraction Collins formula. The light intensity distribution properties of beams focused by a thin lens are further analytically investigated. Results show that the proposed beam exhibits extraordinary propagation properties such as self-splitting and self-shaping, and these transmission properties are closely related to the properties of the correlation function. The cosine-function factor of the total SDOC is responsible for the self-splitting behavior and the Lorentz-function factor determines the self-shaping phenomenon. It is clearly shown that modulating the non-conventional SDOC of a partially coherent beam can alter the coherence length and the degree of nonuniformity, and thus provides an effective way to manipulate its focusing properties. Therefore, the results provide an alternative method for realizing four square beam spots, and have important application prospect in the engineering field.