# The Davenport–Heilbronn twin's exact reflection identity

This note makes the analytic symmetry used by the PP296 contour and folded-zero arguments explicit. It is a derivation from the standard functional equation for primitive Dirichlet L-functions, not a new special-function result or a claim about the Riemann hypothesis. See [NIST DLMF §25.15](https://dlmf.nist.gov/25.15) for the primitive-character functional equation and [Bombieri–Ghosh, *Around the Davenport–Heilbronn function*](https://www.mathnet.ru/links/fb07a8bf9f563e98b864b26a63fa403b/rm9410_eng.pdf) for the function's context.

Let χ be the primitive odd character modulo 5 with χ(1)=1, χ(2)=i, χ(3)=−i, χ(4)=−1. Its Gauss sum is

\[
\tau(\chi)=\sum_{a=1}^{4}\chi(a)e^{2\pi ia/5}
           =-2\sin(\pi/5)+2i\sin(2\pi/5).
\]

For the odd-character completion

\[
\Lambda(s,\chi)=(5/\pi)^{(s+1)/2}\Gamma((s+1)/2)L(s,\chi),
\]

the root number is

\[
\varepsilon_\chi=\frac{\tau(\chi)}{i\sqrt5}
=\frac{2\sin(2\pi/5)}{\sqrt5}
 +i\frac{2\sin(\pi/5)}{\sqrt5}.
\]

It has modulus one and lies in the first quadrant. With φ=(1+√5)/2, its imaginary-to-real ratio is
\[
\frac{\sin(\pi/5)}{\sin(2\pi/5)}=\frac1{2\cos(\pi/5)}=\frac1{\varphi}.
\]

The parameter κ=(√(10−2√5)−2)/(√5−1) is the positive root of κ²+2φκ−1=0. Thus
\[
\frac{1+i\kappa}{1-i\kappa}
\]
also lies on the unit circle in the first quadrant and has imaginary-to-real ratio \(2\kappa/(1-\kappa^2)=1/\varphi\). Hence it equals \(\varepsilon_\chi\) exactly.

Put A=(1−iκ)/2 and B=(1+iκ)/2. Then B/A=εχ, so A εχ=B and B εχbar=A. The standard functional equations
\[
\Lambda(s,\chi)=\varepsilon_\chi\Lambda(1-s,\bar\chi),\qquad
\Lambda(s,\bar\chi)=\bar\varepsilon_\chi\Lambda(1-s,\chi)
\]
therefore give
\[
F(s)=(5/\pi)^{-1/2}\bigl(A\Lambda(s,\chi)+B\Lambda(s,\bar\chi)\bigr)=F(1-s).
\]
This \(F\) is exactly the PP296 normalization \(F(s)=(5/\pi)^{s/2}\Gamma((s+1)/2)f(s)\). Indeed, the character combination \(A\chi+B\bar\chi\) has period-five coefficients \((1,\kappa,-\kappa,-1,0)\), so
\[
f(s)=5^{-s}\sum_{r=1}^{4}a_r\zeta(s,r/5).
\]
The four Hurwitz poles at s=1 cancel because \(\sum a_r=0\), so \(f\) is entire. Separately, each odd Dirichlet L-function has trivial zeros at negative odd integers that cancel the gamma poles in its completion; hence the completed functions and \(F\) are entire. The gamma factor used on the PP296 contours is holomorphic and nonzero there. The identity is exact, while a numerical check of F(s)=F(1−s) alone would not establish it.

For real s>0, grouping the coefficient series into full periods gives
\[
f(s)=\sum_{m\ge0}\left[(5m+1)^{-s}-(5m+4)^{-s}
 +\kappa\bigl((5m+2)^{-s}-(5m+3)^{-s}\bigr)\right]>0.
\]
Each bracket is positive, and the grouped series converges for s>0. The real gamma and exponential prefactors are positive there. Reflection extends \(F(s)>0\) to every real s, so the folded-zero sum omits no real zeros. This conclusion applies to this Davenport–Heilbronn twin and supplies one analytic premise of PP296; the independent contour counts remain a separate task.