Riemann's original paper Ueber Über die Anzahl der Primzahlen unter einer gegebenen Grösse (On the Number of Primes Less Than a Given Magnitude), 1859, is definitely a master well worth reading. In just 8 or so pages he shows how useful the zeta function is for questions about the primes, proves the functional equation, the explicit formula, and makes several deep and far-reaching conjectures (all proven except one infamous example).
This is the paper which (arguably) began the extremely fruitful method of applying complex analysis to number theoretic questions. It lacks details in some places, but it contains a lot of invaluable motivation and exposition.
It certainly helped me to understand why complex analysis is so useful, and how one might discover these connections for himself.
EDIT: Just so you have no excuse, here's a link to an English translation: http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf (Remember that he writes $tt$ for $t^2$ and $\Pi(s-1)$ for $\Gamma(s)$).
