A popular question these days, and, as noted by many above, hard to answer without knowing your local code. But that won't stop me!
1) Your lumber yard can tell you,
2) your engineer can tell you, and
3) I can pull #s out of ..... :
I think your rafters may be a little light, maybe 2x8s at 24"oc instead. Fool around with: http://www.awc.org/calculators/span/...rcalcstyle.asp
Then I thumb through http://www.awc.org/pdf/WSDD/TOC.pdf
and that leads me to lean towards a 6x12, but I may be wrong, often am.
When I can't remember a thing about the WSDD book above, I review my primers at http://bct.nrc.umass.edu/index.php/p...ers-and-beams/
, which is all just a good reminder why options 1 and 2 above are far better ways to go.
It really is very local. My calcs use something called a snow load, which seems inherently silly in LA.
Ridge Beam or Ridge Board? I assume ridge board, which expects a lot more of your beam in question. If it's all hanging off the house, then maybe your 6x10.
And, if you put enough detail into describing what you are doing, you might even get a bite from an actual engineer who sometimes posts to the forums!