Do it on a drill press if possible and do go slow. 300 series stainless is Austenitic which means it will work harden if you try to go too fast or take too big a bite. Best would be a drill with a mechanically controlled down feed.
Now you do want to be cutting a chip so if you go toooo slow , the bit just sits there working the metal and it gets harder.
If I were you I would slowly drill a small pilot hole then go in with the larger bit.
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