All Questions

Let $F$ be a real quadratic field and let $E/F$ be an elliptic curve with conductor 1 (i.e. with good reduction everywhere; these things can and do exist) (perhaps also I should assume E has no CM, ...

Hello, Are there any examples of varieties which are not Shimura varieties or abelian varieties and whose L-functions have been shown to be a product of automorphic L-functions? Thanks. N