Why are $L$-functions a big deal? I've been studying modular forms this semester and we did a lot of calculations of $L$-functions, e.g. $L$-functions of Dirichlet-characters and $L$-functions of cusp-forms. 
But I somehow don't see, why they are considered a big deal. To me it sounds more like a fun fact to say: "You know, the Riemann-$\zeta$ has analytic continuation.", and I don't even know what to say about $L$-functions of cusp-forms. 
So why are $L$-functions such a big thing in automorphic forms and analytic number theory? 
Thanks!
 A: There's a lot one could say, but I'll try to be brief.  Roughly the idea (just like with the zeta functions) is that L-functions provide a way to analytically study arithmetic objects.  Specifically a lot of interesting data is encoded in the location of the zeroes and poles of L-functions, and because L-functions are analytic objects, you can now use analysis to study arithmetic.  Here are some examples:

*

*The fact that $\zeta(s)$ has a pole at $s=1$ implies the infinitude of primes.


*(added) The Riemann hypotheses and generalizations, which are about location of nontrivial zeroes of zeta-/L-functions, have lots of implications, such as refined information about distribution of prime numbers.


*The fact that Dirichlet L-functions do not have a zero at $s=1$ implies there are infinitely many primes in arithmetic progressions.  Dirichlet introduced the notion of L-functions to prove this fact.


*If $E : y^2 = x^3+ax+b$ is an elliptic curve and its $L$-function $L(s,E)$ (which is also the $L$-function of an elliptic curve) is nonzero at the central value $s=1$, then $y^2=x^3+ax+b$ has only finitely many rational solutions.  This is the known direction of the Birch and Swinnerton-Dyer conjecture.


*(added) In addition to knowing just locations of zeroes and poles of L-functions, the actual values of L-functions at special points contain further arithmetic information.  For instance, if $\chi_K$ is the quadratic Dirichlet character associated to an imaginary quadratic field $K$, then the class number formula says $L(1,\chi_K)$ is essentially the class number of $K$.  Similarly, the value of $L(1,E)$ in the previous example  is conjecturally expressed in terms of the size of the Tate-Shafarevich group of $E$ and the number of rational points on $E$.
As mentioned in the comments, $L$-functions are also a convenient tool to associate different kinds of objects to each other, e.g., elliptic curves and modular forms, but are not strictly needed to do this.
Nice $L$-functions will have at least meromorphic continuation to $\mathbb C$, Euler products, and certain bounds on their growth.  For instance, L-functions of eigencusp forms and Dirichlet L-functions.  These properties make $L$-functions nice analytic objects to work with.  In particular, the Euler product provides a way to study global objects from local data (one finite set of data for each prime number $p$).
(added) See also this MathOverflow question.
