TITLE: Why are functional equations important?
QUESTION [39 upvotes]: People who talk about things like modular forms and zeta functions put a lot of emphasis on the existence and form of functional equations, but I've never seen them used as anything other than a technical tool.  Is there a conceptual reason we want these functional equations around?  Have I just not seen enough of the theory?

REPLY [7 votes]: It seems worth mentioning another application of the functional equations.  To fix ideas, I will suppose that $E$ is an elliptic curve over ${\mathbb Q}$, and let $L(E,s)$ be the $L$-function of $E$.  There is then a functional equation relating $L(E,s)$ and $L(E,2-s)$;
one feature of this functional equation is that it has a sign, i.e. has the form
$L(E,s-2) = \pm (\text{ a positive constant })L(E,s),$ where $\pm$ is some fixed sign, depending on $E$, and easily computed (say using the modular form corresponding to $E$).
Thus one find that the order of vanishing of $L(E,s)$ at $s = 1$ is odd (even) precisely
if the sign is $-1$ ($+1$).  
Since the order of vanishing is conjectured to coincide with the rank of $E({\mathbb Q})$
(the BSD conjecture), this is pretty important arithmetic information which is obtained pretty easily from the functional equation.  In particular, if the sign is $-1$, we always expect there to be a rational point of infinite order, and arranging things so that the sign is $-1$ (by twisting by a well-chosen character, say) is a common way of forcing the existence of rational points in various situations.  (There is a large body of research focused around this expected relationship between signs of functional equations and the
existence of rational points on elliptic curves; for example, it lies at the heart of the study of Heegner points by Gross and Zagier and Kolyvagin. Some important recent contributions are by Nekovar, Mazur and Rubin, and T. and V. Dokchitser.)