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An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.

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Nonvanishing of central L-values of quadratic twists?

I suggest you try asking Dipendra Prasad. I have some vague memory that he once told me that (conjecturally) if pi is not self dual then the central value will never vanish.
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