In [Non-vanishing of L-series of modular forms (easy case?)](https://mathoverflow.net/questions/60326) it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is normalized so that the center of the critical strip is given by $s=k/2$.) In particular, for such modular forms, their associated $p$-adic $L$-functions are non-zero. As far as I know the non-vanishing of $p$-adic $L$-functions in the weight 2 case is a highly non-trivial result and relies upon a non-vanishing theorem of Rohrlich on twisted $L$-values. Further, from the non-vanishing of the $p$-adic $L$-function, one can deduce that $L(f,\chi,j)$ is non-zero for all but finitely many pairs $(\chi,j)$ where $\chi$ is a Dirichlet character of $p$-power conductor and $j$ is an integer between $1$ and $k-1$, as long as $p$ is an ordinary prime for $f$.


My questions:


1) Is there a direct argument to prove the non-vanishing of $L(f,\chi,j)$ for all but finitely many $\chi$ and $j$ in the ordinary and weight greater than 2 case (which doesn't use $p$-adic $L$-functions).


2) Is this result known in the non-ordinary case?



Non-vanishing of p-adic L-functions
So, I just talked to David and he pointed me to his paper *L-functions and Division Towers* ([MR0958262](http://www.ams.org/mathscinet-getitem?mr=958262)) whose Theorem 1 is the result (and proof) you're looking for. The proof doesn't care whether $p$ is good or bad or whatever. This takes care of the even weight case, the odd weight case having been dealt with in my comment above. (Also, in case you or anyone reading this were interested in the central point for odd weights, David said he had no idea how to do that).


