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ABSTRACT The Central Molecular Zone (CMZ; the central ∼500 pc of the Galaxy) is a kinematically unusual environment relative to the Galactic disc, with high-velocity dispersions and a steep size–linewidth relation of the molecular clouds. In addition, the CMZ region has a significantly lower star formation rate (SFR) than expected by its large amount of dense gas. An important factor in explaining the low SFR is the turbulent state of the star-forming gas, which seems to be dominated by rotational modes. However, the turbulence driving mechanism remains unclear. In this work, we investigate how the Galactic gravitational potential affects the turbulence in CMZ clouds. We focus on the CMZ cloud G0.253+0.016 (‘the Brick’), which is very quiescent and unlikely to be kinematically dominated by stellar feedback. We demonstrate that several kinematic properties of the Brick arise naturally in a cloud-scale hydrodynamics simulation, that takes into account the Galactic gravitational potential. These properties include the line-of-sight velocity distribution, the steepened size–linewidth relation, and the predominantly solenoidal nature of the turbulence. Within the simulation, these properties result from the Galactic shear in combination with the cloud’s gravitational collapse. This is a strong indication that the Galactic gravitational potential plays a crucial role in shaping the CMZ gas kinematics, and is a major contributor to suppressing the SFR, by inducing predominantly solenoidal turbulent modes.