This is a (far too long) comment on Buzzard's comment about Hida's remark.

I have a guess what Hida must have been saying. There is a result of Shimura that says that the Galois group acts nicely on the central values (in fact any critical value) L-function of eigenforms. In particular if one of them is zero then all the Galois twists are also zero and hence their sum is also zero. Now, even though it may be difficult to show that an L-function doesn't vanish at the centre, it is often easy to show that the sum of the central values of L-functions in a family is non-zero (see, for example, the work of Rohrlich and Rodriguez-Villegas on non-vanishing of L-functions of Hecke characters). 

In the case in question, Maeda's conjecture will imply that if one L-value is zero then the sum of all the L-values must be zero and a contradiction will ensue using (I suspect) Kohnen-Zagier formula. Therefore, no such L-function can vanish. This is a long-standing conjecture in its own right.