I see you already have the starter artifacts, so all you can do is use their history to try to predict the future schedule and location: https://alienwarearena.com/ucf/show/2167784/

Artifacts should rotate and appear on Market for a limited time

getting a degree in archaeology

Check the marketplace where you got the starter ones every new month, and sometimes there'll be one (price is around 1k for these, but they're worth it!). Also play the Steam community events, there's one that can show up there as a reward.