13K miles, I think the dealer will replace your bat if it is indeed dead. I have heard people getting dead battery replace free of charge if the battery died sooner then its expected life (your case seems to fit the bill). So it does not hurt to take it to dealer and let them run a few test to check the alternator as well and see what they say about it. And be nice to the dealer. Now if you killed the bat because you left the light on, then I don't think dealer should give you a free new bat.
