Magic the Gathering question about +1/+1 and -1/-1 counters.?

Basically a question about Persist. If I have Kitchen Finks in play with a -1/-1 counter on it, and I use Ajani Goldmane's second ability to put a +1/+1 counter on it do the counters negate each other? Does it no longer have any counters at all so Persist could activate again, or does it still have both counters on it so Persist wouldn't activate again. I would think it would still have both counters and it wouldn't activate again if Kitchen Finks got buried, but I could be wrong and couldn't find a sure answer.