$s$ in some right half-plane. For almost all (all but finitely many) $v$, we should have $L_v(s) = (p_v(q_v^{-s}))^{-1}$ where $q_v$ is the size of the residue field of $F_v$ and $p_v$ a polynomial of a fixed degree $n$, which is the degree of the $L$-function.

Prototypical $L$-functions of degree $n$ are $L$-functions $L(s, \rho) = \prod L(s, \rho_v)$ of $n$-dimensional Galois representations $\rho : \operatorname{Gal}(\bar F/F) \to \operatorname{GL}_n({\mathbb C})$ (or into $\operatorname{GL}_n(\overline{{\mathbb Q}_p})$) and $L$-functions $L(s,\pi) = \prod L(s, \pi_v)$ of automorphic representations $\pi$ of $\operatorname{GL}_n({\mathbb A}_F)$. In fact it is conjectured that all (nice) $L$-functions are automorphic. These $L$-functions are local-global objects, and one can ask how many local factors $L_v(s)$ determine $L(s)$.

First consider the automorphic case: suppose $\pi, \pi'$ are irreducible cuspidal automorphic representations of $\operatorname{GL}_n({\mathbb A}_F)$, $S$ is a set of places of $F$ and we know that $L(s, \pi_v) = L(s, \pi'_v)$ for all $v \not \in S$. Strong multiplicity one says that if $S$ is finite, then $L(s, \pi) = L(s, \pi')$ (in fact, $\pi \simeq \pi'$). Ramakrishnan [@ramakrishnan:motives] conjectured that if $S$ has density $< \frac 1{2n^2}$, then $L(s, \pi) = L(s, \pi')$, and this density bound would be sharp. This is 