Classical Lower Bound of L(1) assuming GRH Let $L(s)$ be an automorophic $L$-function with conductor $C$ defined by Iwaniec and Sarnak.
What's the classical lower bound of $L(1)$ assuming Riemann Hypothesis?
And what's the reference?
Is that $L(1)\gg \frac{1}{\log C}$ with effectively computed constant?
 A: I'll assume that the degree of the $L$-function is fixed, and that the conductor is going to infinity.  (If the degree is growing, then it is not clear what the right notion of analytic conductor is, and one needs to be careful in using results from the literature.) 
A classical argument of Littlewood shows that on GRH the $L$-value at $1$ is bounded below  by a constant times the Euler product up to $(\log C)^2$.  If now you are also willing to assume the Ramanujan conjecture then it follows that $L(1) \gg (\log \log C)^{-n}$, where $n$ is the degree of the $L$-function.  This argument is classical; for a completely explicit version of it for Dirichlet $L$-functions see the recent paper of Lamzouri, Li and Soundararajan http://arxiv.org/abs/1309.3595 .
If you don't assume the Ramanujan conjecture in addition to GRH then things can be tricky.  It's difficult to get good upper or lower bounds, even on GRH.  The matter is discussed in a paper of Xiannan Li: http://arxiv.org/abs/0904.0850 which appeared in IMRN. 
One more remark:  If there are no Siegel zeros then essentially one can bound the $L$-value at one by taking the Euler product up to a power of the conductor.  The work of Hoffstein and Ramakrishnan (IMRN 1995) essentially shows that Siegel zeros appear only for quadratic  Dirichlet $L$-functions, and so for automorphic forms with no Siegel zeros and satisfying Ramanujan  one would be able to prove a lower bound of $(\log C)^{-n}$.  For a discussion of this in the context of symmetric square $L$-functions, see the paper of Holowinsky and Soundararajan (http://arxiv.org/pdf/0809.1636v1.pdf ).
