We report cross sections for low-energy elastic electron collisions with the diazabenzene molecule pyrazine, obtained from first-principles calculations. The integral elastic cross section exhibits three sharp peaks that are nominally shape resonances associated with trapping in the vacant ${\ensuremath{\pi}}^{*}$ molecular orbitals. Although the two lowest-energy resonances do in fact prove to be nearly pure single-channel shape resonances, the third contains a considerable admixture of core-excited character, and accounting for this channel coupling effect is essential to obtaining an accurate resonance energy. Such resonant channel coupling has implications for electron interactions with the DNA bases, especially the pyrimidine bases for which pyrazine is a close analog. In the absence of data on pyrazine itself, we compare our elastic differential cross section to measurements on benzene and find close agreement.