Selberg Class- Ramanujan Conjecture

Question

The wikipedia for Selberg class L-functions (https://en.wikipedia.org/wiki/Selberg_class) states 4 conditions:

1. Analyticity,
2. Ramanujan conjecture,
3. Functional equation,
4. Euler product.

I would like to get a better intuitive understanding of the Ramanujan conjecture and why it is included as a condition. What properties of l-functions that violate RC but abide by the other 3 preclude RH?

Answer 1

I've seen a lot of Dirichlet series, but I've never seen a Dirichlet series with a functional equation of the shape $s \mapsto 1-s$, an Euler product, meromorphic continuation, and no Ramanujan-Petersson conjecture... sort of.

Broadly speaking, everything that people call an $L$-function satisfies all four. We also expect that every $L$-function comes from either an automorphic form on some algebraic group, a motive, or a Galois representation (actually, we expect that every $L$-function comes from all three, one piece of the Langlands program).

For automorphic forms on $\mathrm{GL}(n, \mathbb{R})$, one can show that the Ramanujan-Petersson conjecture holds at least on average in the sense that $$ \sum_{n \leq X} a(n)^2 \sim c X, $$ where $a(n)$ are the coefficients of the $L$-function. (I note that normalizing the functional equation to have shape $s \mapsto 1-s$ also has the effect of normalizing the coefficients). We don't actually know if the Ramanujan-Petersson conjecture holds for these $L$-functions, though we expect that it does.

Answer 2

Try

$$F(s)=L(s+\delta,\chi_4)L(s-\delta,\chi_4)$$ where $L(s,\chi_4) = \sum_{n\ge 1}\chi_4(n)n^{-s},\chi_4(2n)=0,\chi_4(2n+1)=(-1)^n$ and $\delta\in (0,1/2)$.

• $F(s)$ extends to an entire function

• It has an Euler product $F(s)=\exp(\sum_{p^k} b_{p^k} p^{-sk})$ with $b_{p^k}=\frac{\chi_4(p^k)}k( p^{\delta k}+p^{-\delta k})$

• As $\Lambda(s)=\Gamma(s/2+1/2)(4/\pi)^{s/2} L(s,\chi_4) = \Lambda(1-s)$ we get the Selberg class functional equation $$\lambda(s)=\Gamma((s+\delta)/2+1/2)\Gamma((s-\delta)/2+1/2)(4/\pi)^s F(s) = \lambda(1-s)$$

• The coefficients $\chi_4(n) n^\delta\sum_{d|n} d^{-2\delta}$ aren't $O(n^\epsilon)$ so $F(s)$ doesn't satisfy the Ramanujan conjecture

• Of course the Riemann hypothesis doesn't hold for $F(s)$.