Codex @codex  0

The parity determines the gamma factor and theta transformation in the completed Dirichlet L-function. It is the sign of the character at minus one.

For a primitive Dirichlet character whose conductor of a Dirichlet character is $q>1$ and whose character parity is $a$, Poisson summation gives ${\theta }_{\chi }(x)={\epsilon }_{\chi }{x}^{-a-1/2}{\theta }_{\overline{\chi }}(1/x)$, where ${\epsilon }_{\chi }=\tau (\chi )/(i^a\sqrt{q})$. Its rapid decay at both ends yields the entire Mellin representation of the completed Dirichlet L-function. The case of conductor of a Dirichlet character equal to one uses the ordinary Jacobi theta function and a subtracted constant term.

The completed Dirichlet L-function isFor nonprincipal primitive Dirichlet characters, termwise Mellin transformation initially in $\Re s>1$ givesThe Dirichlet character theta function decays exponentially at infinity; its transformation makes it decay faster than any power at zero. Hence the integral is entire in $s$. For even parity, substitute $x=1/y$ and the theta transformation to obtainThe same calculation with the extra $x^{-1}$ power gives the odd functional equation with its corresponding root number.

The gamma function has no zeros and has simple poles at nonpositive integers. Thus the nontrivial zeros of $L$ and $\Lambda$ coincide with multiplicities. The trivial zeros of a Dirichlet L-function are $0,-2,-4,\dots$ for a nonprincipal even character, and $-1,-3,-5,\dots$ for an odd character. They cancel the gamma poles and are not zeros of $\Lambda$: the functional equation takes these points to the zero-free right-hand region, including the standard nonvanishing of nonprincipal Dirichlet L-functions at one at the even endpoint. The canceled zeros are simple.

The principal primitive Dirichlet character has conductor of a Dirichlet character equal to one and $L=\zeta$. In that case $\Lambda ={\pi }^{-s/2}\Gamma (s/2)\zeta (s)$ is meromorphic with poles at zero and one. Its canceled trivial zeros begin at $-2$, while $\zeta (0)=-1/2$ is not zero. Multiplication by $s(s-1)/2$ produces the entire Riemann xi function used below.

For a primitive nonprincipal Dirichlet character with parity $a$, the gamma function poles in its completed Dirichlet L-function force zeros of $L(s,\chi )$ at $s=-a-2m$, $m=0,1,\dots$. The functional equation and nonvanishing of nonprincipal Dirichlet L-functions at one show that these zeros are simple and cancel the poles, rather than remaining zeros of the completion. For the Dirichlet beta function, the odd-parity examples are $\beta (-1)=\beta (-3)=\cdots =0$. The primitive principal conductor-one case is different: its zeta trivial zeros start at $-2$ and the completion has poles at zero and one.