Axiom 4: Euler Product

$$F(s)=\prod_p F_p(s),$$

where the product is over the rational primes. Here

$$F_p(s)=\exp\left(\sum_{k=1}^\infty \frac{b_{p^k}}{p^{ks }} \right)$$

with $b_n\ll n^\theta$ for some $\theta<\frac12$.

Note that this implies $a_1=1$, so $F(s)=1$ is the only constant function in the Selberg class.

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