By using slowly-varying envelope approximation, a family of solutions of the paraxial wave equation are found in theory, which represents a new family of ultrashort pulsed beams called ultrashort pulsed complex argument Sinc-Gaussian beams. These pulsed beams with a certain frequency have a nearly complex argument Gaussian profile, a sinc function temporal shape. The ultrashort pulsed complex argument Sinc-Gaussian beams and their propagation properties in free space are studied detailedly, such as intensity on the axis, transversal intensity distribution of the pulsed beams, polarity reversal, pulse time delay, etc..