The spin dynamics of copper pyrazine dinitrate (Cu(C₄H ₄N₂)(NO₃)₂), a model spin-1/2 Heisenberg antiferromagnetic (AF) chain system, was investigated by means of electron spin resonance (ESR). Using the high-field ESR we evidenced the inequivalence of Cu sites belonging to adjacent spin chains in the ac planes of this compound. It was revealed that the dominating interchain interaction is of zig-zag-type. This interaction gives rise to geometrical frustration strongly affecting the character of AF ordering. Combining our experimental findings with the results of a quasiclassical approach we predict that at low temperatures the system orders in an incommensurate spiral state.