Let $\displaystyle F(s) = \zeta(s)L(s,\chi_1)L(s,\chi_2)L(s,\chi_1\chi_2)$, for two real primitive characters $\displaystyle \chi_1, \chi_2$. Multiplication of Euler products of these functions gives us

$\displaystyle F(s) = \displaystyle \sum_{n=1}^\infty a_n n^{-s}$

with $\displaystyle a_1=1$. Davenport (in p.129 of Mult. Numb. Theory) claims that taking logarithms of Euler products gives

$\displaystyle \displaystyle \log F(s) = \sum_p \sum_{m=1}^\infty \big(1+\chi_1(p^m)\big)\big(1+\chi_2(p^m)\big)m^{-1} p^{-ms},$

which is fine, but then claims that since the coefficients here are nonnegative, we have $\displaystyle a_n \geqslant 0$ for all $\displaystyle n$. Can someone show me the reasoning for this last fact? It doesn't seem obvious to me.