This paper is the first in a series of papers describing the impact of antenna instrumental artefacts on the 21-cm cosmology experiments to be carried out by the low frequency instrument (SKA1-LOW) of the Square Kilometre Array telescope (SKA), i.e., the Cosmic Dawn (CD) and the Epoch of Reionization (EoR). The smoothness of the passband response of the current log-periodic antenna being developed for the SKA1-LOW is analyzed using numerical electromagnetic simulations. The amplitude variations over the frequency range are characterized using low-order polynomials defined locally, in order to study the impact of the passband smoothness in the instrument calibration and CD/EoR Science. A solution is offered to correct a fast ripple found at 60~MHz during a test campaign at the SKA site at the Murchison Radio-astronomy Observatory, Western Australia in September 2015 with a minor impact on the telescope's performance and design. A comparison with the Hydrogen Epoch of Reionization Array antenna is also shown demonstrating the potential use of the SKA1-LOW antenna for the Delay Spectrum technique to detect the EoR.