It depends on its manufacturer; I have two different brands of GFCI's installed in my home, one made by Laviton (I actually have two different designs by Leviton / both with indicator lights) and one by Pass & Seymour.
Laviton’s GFCI has an indicator light that stays green when properly wired and active (power ON), the light goes OFF as soon as the power goes OFF / you push test button.
Pass & Seymour’s indicator light (at least in my model) stays OFF when the GFCI is functional and turns red when you push test button / the power goes OFF.
Check the brand, go to their website and read the manual, it should state the features / behavior of the indicator. I remember seeing GFCI with light changing color from green to red / orange as well, I’m just not sure of the brand.