Always ask for what you're worth. I do my research with sites like glassdoor.com, and work from there.
You cannot get paid money that you do not ask for, so why not ask? The worst case scenario is that they say no, and more often than not, this will come with a counter-offer.
You'll find as you move higher up the professional ladder, salary negotiations become true negotiations. If they can't afford to pay you what you're asking for, or they come back with a counter-offer that is below your asking price, see what concessions you can get. An extra week of vacation, a certain number of guaranteed telecommuting days a week, higher matching 401(k) contributions, or a guaranteed raise after a specific period of time are all be big benefits that me or my colleagues have successfully negotiated for in the past.
Whatever happens, be sure to get their offer (and concessions, if any) in writing, and take a day or two to mull it over if it's not exactly what you're looking to get. That gives you some time to make sure that you're taking a position with compensation that won't leave you bitter, and it also gives them time to sweat it out and possibly up their offer if they think you might turn it down.