Selberg class axioms (reviewed)

The ** Selberg axioms** for $F(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$ in the Selberg class are:

1. Analyticity: $(s-1)^mF(s)$ is an entire function of finite order for some non-negative integer $m$
2. Ramanujan Hypothesis: $a_n = O_{\epsilon}(n^\epsilon)$ for any fixed $\epsilon>0$
3. Functional equation: there is a function $\gamma_F(s)$ of the form $$\gamma_F(s)=\epsilon Q^s \prod_{i=1}^k\Gamma(\lambda_is+\mu_i)$$ where $|\epsilon|=1$, $Q>0$, $\lambda_i>0$, and Re$(\mu_i)\ge 0$ such that $$\Lambda(s)=\gamma_F(s)F(s)$$ satisfies $$\Lambda(s)=\overline{\Lambda}(1-s)$$ where $\overline{\Lambda}(s)=\overline{\Lambda(\overline{s})}$.
4. Euler product: $a_1=1$, and $$\log F(s)=\sum_{n=1}^\infty \frac{b_n}{n^s}$$ where $b_n=0$ unless $n$ is a positive power of a prime and $b_n\ll n^\theta$ for some $\theta<1/2$.