My argument is very simple. In almost all the nations where the death penalty was recently abolished, we have witnessed a sharp upswing in the crime rate. Therefore, IMO, the death penalty must be there to deter these sort of crimes.
I don't think it's the death penalty per se that's the deterrent. I think it's more the prevailing attitude of particular societies and their lawmakers. If they have something like the death penalty, they are perceived as tough and no-nonsense. If they don't have it, they look passive.
That may be one of the reasons. For example, Sweden is having one of the highest rape rates in the world, fueled by the lax punishments given to the perpetrators. Most of the time, the rapists avoid jail time by just paying a monetary fine, or doing community service.