In this work we study the solvatochromic shift of a selected low-energy excited state of alizarin in water by using a linear-scaling implementation of large-scale time-dependent density functional theory (TDDFT). While alizarin, a small organic dye, is chosen as a simple example of solute-solvent interactions, the findings presented here have wider ramifications for the realistic modeling of dyes, paints, and pigment-protein complexes. We find that about 380 molecules of explicit water need to be considered in order to yield an accurate representation of the solute-solvent interaction and a reliable solvatochromic shift. By using a novel method of constraining the TDDFT excitation vector, we confirm that the origin of the slow convergence of the solvatochromic shift with system size is due to two different effects. The first factor is a strong redshift of the excitation due to an explicit delocalization of a small fraction of the electron and the hole from the alizarin onto the water, which is mainly confined to within a distance of 7 Å from the alizarin molecule. The second factor can be identified as long-range electrostatic influences of water molecules beyond the 7 Å region on the ground-state properties of alizarin. We also show that these electrostatic influences are not well reproduced by a QM/MM model, suggesting that full QM studies of relatively large systems may be necessary in order to obtain reliable results.