Lone pigeon in garden

A week ago a pigeon came to our small city garden and has not left since. It either sits on our patio all day or in our next door neighbours low guttering. It has no ring on its leg but seems very tame as it does not moved when I walk up to it. What should I do? I am tempted to feed it but I'm not sure that is correct. I feed the small birds in a feeder in the tree and on a bird feeder but this pigeon does not seem to be hungry to pick up the food the other birds scatter on the ground. Please advise me.