I went for a short ride today and twice when I started opening the throttle in 1st or 2nd gear (can't remember) the bike just hesitated... What's that all about?
Warming it up could be part of the issue, but it sounds to me it is the normal "carb fart", and needs the typical Stage 1 , or at least the pilot jet swapped to a 45.
I have a 2006 1200L carb. i recently had similar issues. i do have a stage one kit already installed. depending on how many miles you have on the bike, >20-30K it sounds like an intake leak. if the bike sputters etc when warm. you can check for this leak by spraying contactcleaner near the intake. if it dies instantly you have the problem. Also i suggest you go to a local harley shop, not a dealer. harley performance shop etc. The dealers are just not good at tuning jets and performance carbs. Goodluck. p.s. carb or not, it should not be constantly hesitating while accelerating. get it tuned by a pro.