Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n > 1 variables and let. be a character of R-x. Let M-i(u) be the number of solutions of f = u in (R/P-i)(n) for i is an element of Z(>= 0) and u is an element of R/P-i. These numbers are related with Igusa's p-adic zeta function Z(f,chi) (s) of f. We explain the connection between the M-i(u) and the smallest real part of a pole of Z(f chi) (s). We also prove that M-i(u) is divisible by q([(n/2)(i-1)]), where the corners indicate that we have to round up. This will imply our main result: Z(f,chi)(s) has no poles with real part less than -n/2. We will also consider arbitrary K-analytic functions f.