Generalized Riemann hypothesis for Selberg class

For a complex variable $s$, let

$$F(s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}$$

be a Dirichlet series with $a_{1}=1$ that converges absolutely for $\operatorname{Re}(s)>1$, and say that $F$ belongs to the Selberg class $\mathcal{S}$ when the following hold.

1. (Analytic continuation) There is an integer $m\ge 0$ such that $(s-1)^{m}F(s)$ extends to an entire function of finite order; equivalently, $F$ continues meromorphically to $\mathbb{C}$ with at most a pole at $s=1$.

2. (Ramanujan bound) For every $\varepsilon>0$, $a_{n}\ll_{\varepsilon}n^{\varepsilon}$.

3. (Functional equation) There exist $Q>0$, an integer $k\ge 0$, and numbers $\omega_{i}>0$, $\mu_{i}\in\mathbb{C}$ with $\operatorname{Re}(\mu_{i})\ge 0$ $(1\le i\le k)$, and $\epsilon\in\mathbb{C}$ with $|\epsilon|=1$, such that with

$$\gamma(s)=Q^{s}\prod_{i=1}^{k}\Gamma(\omega_{i}s+\mu_{i}),\qquad \Phi(s)=\gamma(s)F(s),$$

one has

$$\Phi(s)=\epsilon\,\overline{\Phi(1-\bar{s})}\qquad\text{for all }s\in\mathbb{C}.$$

4. (Euler product) $F(s)\neq 0$ for $\operatorname{Re}(s)>1$ and

$$\log F(s)=\sum_{n=1}^{\infty}\frac{b_{n}}{n^{s}}\qquad(\operatorname{Re}(s)>1),$$

where $b_{n}=0$ unless $n=p^{\nu}$ is a prime power, and $b_{n}\ll n^{\theta}$ for some $\theta<\tfrac{1}{2}$.

Then for every $F\in\mathcal{S}$ and every $s\in\mathbb{C}$,

$$F(s)=0\ \text{ and }\ 0<\operatorname{Re}(s)<1\quad\Longrightarrow\quad \operatorname{Re}(s)=\tfrac{1}{2}.$$

The zeros of $F$ outside the strip $0\le\operatorname{Re}(s)\le 1$, which occur only at the poles $s=-(\mu_{i}+n)/\omega_{i}$, $n\in\mathbb{Z}_{\ge 0}$, of the factors of $\gamma$, are excluded from the assertion.