M-DNA is a type of metalated DNA that forms at high pH and in the presence of Zn, Ni, and Co, with the metals placed in between each base pair, as in G-Zn-C. Experiments have found that M-DNA could be a promising candidate for a variety of nanotechnological applications, as it is speculated that the metal d-states enhance the conductivity, but controversy still clouds these findings. In this paper, we carry out a comprehensive ab initio study of eight G-Zn-C models in the gas phase to help discern the structure and electronic properties of Zn-DNA. Specifically, we study whether a model prefers to be planar and has electronic properties that correlate with Zn-DNA having a metallic-like conductivity. Out of all the studied models, there is only one which preserves its planarity upon full geometry optimization. Nevertheless, starting from this model, one can deduce a parallel Zn-DNA architecture only. This duplex would contain the imino proton, in contrast to what has been proposed experimentally. Among the nonplanar models, there is one that requires less than 8 kcal/mol to flatten (both in gas and solvent conditions), and we propose that it is a plausible model for building an antiparallel duplex. In this duplex, the imino proton would be replaced by Zn, in accordance with experimental models. Neither planar nor nonplanar models have electronic properties that correlate with Zn-DNA having a metallic-like conductivity due to Zn d-states. To understand whether density functional theory (DFT) can describe appropriately the electronic properties of M-DNAs, we have investigated the electronic properties of G-Co-C base pairs. We have found that when self-interaction corrections (SIC) are not included the HOMO state contains Co d-levels, whereas these levels are moved below the HOMO state when SIC are considered. This result indicates that caution should be exercised when studying the electronic properties of M-DNAs with functionals that do not account for strong electronic correlations.