Critical line and critical strip

The critical line is the line of symmetry in the functional equation of the $L$-function. In the usual normalization the functional equation associates $s$ to $1-s$, so the critical line is $\sigma=\frac12$.

In the usual normalization the Dirichlet series and the Euler product converge absolutely for The functional equation maps $\sigma>1$ to $\sigma<0$. The remaining region, $0<\sigma<1$ is known as the critical strip.

By the Euler product there are no zeros in $\sigma>1$, and by the functional equation there are only trivial zeros in $\sigma<0$. So all of the nontrivial zeros are in the critical strip, and the Riemann Hypothesis asserts that the nontrivial zeros are actually on the critical line.

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