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So -- long story short -- I share Nick's intuition that the L-function is the entire family of quadratic twists, and I went so far as to cook up a theory of L-groups for (split, for now) metaplectic groups in order to understand this better.
When $G$ is the simplest metaplectic group $\widetilde{SL}_2$, I argue that the L-group ${}^L G$ is a group scheme over $\mathbb{Z}$, which is isomorphic to $SL_2 \times \Gamma$ ($\Gamma$ the Galois group), but noncanonically and not until base change to $\mathbb{ZZ}[i]$. However, each additive character of ${\mathbb A} / {\mathbb Q}$ gives such an isomorphism, and this realizes a bijection between quadratic twists and (L-)isomorphisms of the L-group to $SL_2 \times \Gamma$. Taking the standard representation of $SL_2$, you get the L-function of a quadratic twist.
Following up on Nick's question -- the Langlands correspondence for metaplectic groups is something I've thought about a lot in the past few years. I have a paper on L-groups for Metaplectic Groups on the ArXiv, for example. Since that paper is hard to read (I'm working hard to get rid of all the Hopf algebra machinery used there), I'll summarize here. An L-function should come from two pieces of data: an automorphic representation of $G$ (for half-integral weight eigenforms, $G$ is a metaplectic group) and an algebraic (finite-dimensional) representation of the L-group ${}^L G$.
Why look at roots instead of coroots? You can see the $F_4$ embedded in $E_6$, for example, by seeing each simple coroot of $F_4$ as a sum of one or two coroots from $E_6$. When one group embeds in another compatibly with max tori, e.g. $F_4$ in $E_6$, the map is covariant from coroot lattice of the small group to coroot lattice of the large group.
I think the general answer is in SGA1, and someone could probably find the positive answer there and give it as an official answer. Coincidentally, I was wondering about this a bit last week, and I found papers of J. Stix in my googling. For example, see Corollary 5.3 in: mathi.uni-heidelberg.de/~stix/preprints/STIXSvK.Juli2005.pdf
A short comment for now: Gauss understood the connection between lattice points on spheres and class numbers of definite quadratic forms: The number of representations of $m$ as a sum of 3 squares is a constant times $h(-m)$ or $h(-4m)$ depending on the congruence of $m$ mod $8$, as I recall. The estimate (a) can probably be deduced from counting lattice points in $R^3$.
Hopefully you'll get some more suggestions below. But I want to remark that giving a student a topic such as "elliptic curves" or "cryptography" is giving way too much breadth. It's up to you to help the student find a narrow enough topic to be manageable. Otherwise you'll get some awful vague pseudo-mathematical papers. E.g., within cryptography, you could have the student write about El Gamal and the discrete log problem. Or with elliptic curves, a student could write about the Hasse bound and state the Sato-Tate conjecture.
