Weil Conjectures for schemes

Question

I have two questions.

1. What class of schemes are the Weil conjectures true for? Projective schemes? Do I need additional hypotheses? I know from my question (The Weil conjecture involving Betti numbers.) that we need a scheme over the Spec of some ring of integers.
2. Can you provide an example of a scheme as in 1. that is not a projective variety?

Thank you in advance.