Context. Star formation in the central molecular zone (CMZ) is suppressed with respect to that of the Galactic disk, and this is likely related to its high turbulent environment. This turbulence impedes the potential detection of prestellar cores. Aims. We present the temperature, density, and spatial structure of the CMZ molecular cloud G+0.693-0.027, which has been proposed to host a prestellar cluster in the Sgr B2 region. Methods. We analysed multiple HC3N rotational transitions that were observed with the IRAM 30m, APEX, Yebes 40m, and GBT radio telescopes, together with SMA+APEX spatially resolved maps. Results. The spectral shape of HC3N lines shows three distinct velocity components: a broad component with a line width of 23 km s(-1) (C1), and two narrow components with line widths of 7.2 and 8.8 km s(-1) (C2 and C3). This suggests that a fraction of the molecular gas in this cloud is undergoing turbulence dissipation. From an initial local thermodynamic equilibrium analysis, we found column densities of N = (6.54 +/- 0.07) x 10(14) cm(-2), (9 +/- 3) x 10(14) cm(-2), and (3.6 +/- 0.7) x 10(13) cm(-2) for C1, C2, and C3, respectively. These values were used as input for a subsequent non-local thermodynamic equilibrium analysis, in which we found H-2 densities of 2 x 10(4) cm(-3), 5 x 10(4) cm(-3), and 4 x 10(5) cm(-3) and kinetic temperatures of 140 K, 30 K, and 80 K for C1, C2, and C3, respectively. The spatially resolved maps confirm that the colder and high-density condensations C2 and C3, which peak in the 70-85 km s(-1) velocity range, have deconvolved sizes of 9 '' (0.36 pc) and 7.6 '' (0.3 pc), respectively, and are embedded in a more diffuse and warmer gas (C1). Conclusions. The larger-scale structure of the Sgr B2 region, consistently with previous works, shows a hole at 40-50 km s(-1) that is likely due to a small cloud that shocked the Sgr B2 region and is spatially related with a massive cloud at 60-80 km s(-1). We propose that the impacting small cloud sequentially triggered the formation of Sgr B2(M), (N), and (S) and the condensations in G+0.693-0.027 during its passage. The two condensations are in a post-shocked environment that has undergone internal fragmentation. Based on the analysis of their masses and the virial parameters, C2 might expand, while C3 might further fragment or collapse.