Without enhancing it first, thus breaking a lot of it, you won't get a relic fragment from loot. Now swoop does drop minor artifacts, some of which can unravel into relic frags.
Honestly, luck is so across the board in it's effectiveness it's hard to say exactly how much is the right amount. Some players swear they get great loot wearing 800 - 1000 luck. Others insist you need 1200+. You can get in the range of 1900-2500 with the 10th anniversary statue.
I dunno, I can't recall for sure, but I don't think they have ever told us exactly how luck of different levels actually affects the loot tables. We all know how it's supposed to work. Luck gives you a chance to bump the number of items, properties and intensity. Detailed formula with exact calculations, I don't think it's ever been put out.