Analytical solutions have been obtained for the electromagnetic scattering by a modified Luneberg lens with the permittivity of arbitrary parabolic function. They are expressed by four spherical vector wave functions for radially stratified medium which were introduced for the Luneberg lens by C. T. Tai. They consist of the confluent hypergeometric function and a "generalized" confluent hypergeometric function, in which the parameters for the permittivity of arbitrary parabolic function are involved. The characteristics of the modified Luneberg lens are numerically investigated using exact solutions in comparison with that of the conventional Luneberg lens. The bistatic cross section, the forward cross section and the radar cross section are studied in detail. The near-field distribution is also investigated in order to study the focal properties of the Luneberg lens. The focal shifts defined by the distance between the geometrical focal point and the electromagnetic focal point are obtained for various ka (k is the wave number and a is the radius of the lens). The focal shift normalized to the radius of the sphere becomes larger as ka is smaller. However it drops down rapidly for ka5 when the peak of the electric field amplitude appears on the surface of sphere.