The Riemann-Siegel theta function $\theta(t)$ is well-behaved and satisfies $\zeta(\frac{1}{2}+it)=Z(t)e^{-i\theta(t)}$ for real $Z(t)$. Because of how smoothly $\theta(t)$ changes, $Z(t)$ changes signs around roots, allowing us to find zeroes on the critical line using Brent's method and similar methods.
I'm trying to find a generalization of the Riemann-Siegal theta function to Dirichlet L-functions that has a similar "explicit formula". The argument of L-functions on the critical line is similarly smooth, but it's distinctly different from the argument of the zeta function shown from "Orientation" of $\zeta$ zeroes on the critical line:
Argument of Dirichlet L-function with character period 17 and index 2
The only evidence I have that such a function exists is from a piece of the source code for finding zeroes of L-functions in Sage, which describes "rotating" values of the L-function to the reals, but I do not understand what it is calculating.
What could a generalization of the Riemann-Siegal theta function be, and how would we calculate it?
Best Answer
For $\chi$ a primitive Dirichlet character, let $$\xi(s,\chi) = \left(\frac{q}{\pi}\right)^{(s+a)/2} \Gamma\left(\frac{s+a}{2}\right) L(s,\chi)$$ with $a=0$ if $\chi$ is even and $a=1$ if $\chi$ is odd, we have the functional equation $$\xi(s,\chi) = \varepsilon(\chi) \xi(1-s,\overline{\chi})$$ where $\varepsilon(\chi) = \frac{\tau(\chi)}{i^a \sqrt{q}}$ has modulus $1$.
For $s=1/2+it,t\in \Bbb{R}$ it becomes $$\varepsilon(\chi)^{-1/2}\xi(1/2+it,\chi) = \overline{\varepsilon(\chi)^{-1/2}\xi(1/2+it,\chi)}$$ ie. the $\Bbb{R\to R}$ function with the same critical line zeros is $$Z(t,\chi)=\varepsilon(\chi)^{-1/2}\xi(1/2+it,\chi)$$