You'll have to see if it persists, but he might just be one of the 20% of babies who are colicky. If so, check out the Happiest Baby on the Block and the 5 S's!
Any face rash? Resistance to eating? Hard belly or gas? Blood or mucus in the diaper? Cottage cheese spit up? These are signs of dairy intolerance. DD1 started showing signs at 14 days including the inconsolable crying. Hopefully it's not your issue, but it gives you something to keep an eye out for.
Might be overly tired. That is usually the problem with our LO. Try gripe water! I really don't know what it does, but it works almost every time! Also, turn off the lights and everything that is noisy and try swaddling and rocking/bouncing.
Mine did that too. I was trying to nurse and he wasn't getting much. Turns out he was starving. I supplemented with 1 oz of formula and he was a total different baby after that. How many wet and dirty diapers is he making? That's how you can tell if he's getting enough milk or not.
Doesn't console when offering a feeding? There are only a handful of reasons why they would be crying. Diaper change, gas (need burp), hungry, needs some love (rocking). Orrrrr colicky but I don't have experience with that. Hope you figure something out. Ps. The baby swing is the holy grail for us. Our LO loves the swing.
Re: Crying/inconsolable newborn