At the start of the last century, the age of consent was 12.
Now it's 16. However, many young people of both sexes think that that they are mature enough and see themselves as individuals and adults at that age- i.e. old enough to make their own decisions, to decide for themselves what they do and do not want to do.
In short, if she feels OK with it, then go for it. As MB says, though, don't have sex as she IS underage and you are technically raping her if you have sex.
Some other points- if girls of 13 are old enough to have babies and bring them up- two of my cousins were pregnant at 13, and are now bringing up the children- then a girl of 15 should be able to have a relationship.
Ina ddition, girls generally mature 3 years ahead of boys, so you've got the same mental age...
But I would advise caustion. Yes, go out, yes, have fun, but don't have full sex until she is over the age of consent. At the maximum you've got a year to wait, and if you love each other, not just lust, you will be able to wait.
Or just don't get caught...