i haven't done any testing, so i can't answer with specifics, but i will say...
Do a lot of visitors perform a search once they get to your site?
i'm sure the answer to this depends heavily on visitor intent and the type of site.
from experience as a site visitor i can tell you that it is often not obvious where the search function is, so this will factor into your "conversion rate" through on-site search. you might also consider the browsing platform of your visitors as the use of search and the visibility of the search box may change significantly for mobile visitors.