The orientation partition function, the associated angle-averaged Mayerf-function, and the effective isotropic potential for a pair of dipoles are expressed in terms of a twice-modified incomplete Struve function of zero order. The partition function is a genus one, order one, type two entire function, with an infinite number of zeros along the imaginary axis. Analytic expressions are derived that characterize it throughout the complex plane. These expressions provide efficient means for computing the partition function, the angle-averaged Mayerf-function, and the effective potential. The partition function's distribution of zeros and classical low-temperature behavior resemble that of the partition function for an Ising dipole pair and for a dipole in an external field, but are strikingly different from that of the Boltzmann factor of the attractive component of the Lennard-Jones effective potential of dipolar thermodynamic perturbation theory.