First-principles band structure calculations were performed for the ternary alkaline-earth silicon nitride Ba5Si2N6 using the LSW method. The calculations show that both the (zero-)dimensionality of the [Si2N6]10- dimeric units present in this structure and the coordination number of nitrogen by silicon have strong influences on the local electronic structure of these atoms. The interaction between the semicore-level states Ba 5p and N 2s is significant. Finally the compound is calculated to be a semiconductor with an indirect gap of 0.7 eV. The top of the valence band is dominated by the 2p states of the N atoms. The conduction bands are determined by N 3s states hybridized with Ba 6s states.