[math-fun] Class number query

Let f(n) = SUM(0 <= k < n; (k^2) mod n) where mod is the remainder function. It looks as if, for prime p >= 5 f(p) = 1 if p == 1 (mod 4) 2c+1 if p == 3 (mod 4) where c is the class number of Q(sqrt(-p)) Is this true? Well known?