In Goldfeld's book on automorphic representations, he computes the local root number of every ramified place of a character $\omega$ to be the Gauss sum $$\sum_{\substack{j=1\\(j,p)=1}}^{p^r}\omega(j)e^{\frac{2\pi ij}{p^r}}$$ Goldfeld mentions that the global root number (which is the multiplication of all the local ones) is again a Gauss sum of the form $$\sum_{\substack{j=1\\(j,n)=1}}^{n}\omega(j)e^{\frac{2\pi ij}{n}}$$ where $n$ is the conductor of $\omega$, however, this doesn't make sense to me.
Consider the simple case of the two principle Gauss sums $$\sum_{\substack{j=1\\(j,p)=1}}^{p}\omega(j)e^{\frac{2\pi ij}{p}},\quad\sum_{\substack{j=1\\(j,q)=1}}^{q}\omega(j)e^{\frac{2\pi ij}{q}}$$ where $\omega$ is now assumed to be ramified only at $p$ and $q$ and have conductor $pq$. Why should the multiplication of these two sums be again a Gauss sum?
Thanks in advance