Lindelof Hypothesis implying Selberg Eigenvalue Conjecture? The Generalized Lindelof Hypothesis says that for the $L$-function of an automorphic form we have
$$L(1/2+it)\ll Q(t)^{\epsilon}$$
for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $t$.
For a Maass form $\phi$ of level $N>1$ and eigenvalue $1/4+\lambda^2$, will Lindelof Hypothesis imply Selberg Eigenvalue conjecture, i.e., $\lambda\geq 0$?
I know that functoriality results such as Gelbart-Jacquet, Shahidi, Kim gives a very good bound, such as Kim-Sarnak bound of 7/64. Obviously functoriality for of all symmetric powers implies the Selberg eigenvalue conjecture. The proof involves some non-vanishing results for families of $L$-functions (Kim-Sarnak, Luo-Rudnick-Sarnak). I am thinking about the relation of the Selberg eigenvalue conjectures and the GRH/Lindelof.
Lastly, GRH itself at a single point for one L-function probably does not say much about the Selberg eigenvalue conjecture because GRH only excludes non-trivial zeros. A violation of Selberg eigenvalue conjecture will give us a zero from Gamma factors (thus it is trivial zero). But family of L-functions with GRH or Lindelof possibly can tells more about non-vanishing (trivial zeros) of L-functions.
 A: To see that this is not the case, consider that it is well known that the Generalized Riemann Hypothesis implies the Lindelöf hypothesis.
On the other hand, not even the full GRH implies the Selberg Eigenvalue conjecture (or any other Generalized Ramanujan-type conjecture).
In fact, the best bound we can prove under GRH for a Maass form $\pi$ is $|\mathrm{Re}\mu_\pi(\infty,j)|\leq 1/4$, which is quite far from the $|\mathrm{Re}\mu_\pi(\infty,j)|=0$ we need.
There's also an argument of Sarnak that goes something like this:
Take $\pi$ a Maass form, and assume we know all the general Ramanujan conjectures for $\pi$ (we know from Langlands that we can treat the eigenvalue conjecture as a Ramanujan conjecture at infinity).
Then we can prove that $L(s,\pi)$ is uniformly bounded in $\mathrm{Re}(s)>1+\epsilon$, with $\epsilon >0$, but we can't push this result even to $\mathrm{Re}(s)=1$.
On the other direction, we have the same problem when we try to use the Generalized Lindelöf hypothesis to give information about $\pi$ beyond $\mathrm{Re}(s)=1$.
