by the way, I think that 不死 means "immortal" in a physical way (meaning "that can't die"). If you would like to refer to a concept, not a living object, you should use 不滅 (which means "that can't be destroyed").
See these two examples.
不死の化け物 -> an immortal monster
不滅の愛 -> an immortal love
a monster is a living thing, "love" is an abstract object.