I'm mostly sure I've got any Burgess zeta function continued to $ \mathbb{C} $.

I don't even do anything close to mathematics for a living, and sharing this is new to me, so please forgive grammatical errors where the concept's still sound and clear. Nothing's peer-reviewed, so take "proven" in that context. The work is pushing 20+ pages, so I'm going to try to give the best quick version I can:

(Proven) Hurwitz zeta functions can be converted into a Riemann zeta function, where $ n \equiv ? \% p $, like this: $ \zeta \left( s, \frac{q}{p} \right) p^{-s} $

(Proven) Using additional imaginary variables, eigenvectors that are congruent over multiplication to multiplication modulo any number may be created. One easy method is by multiplying cis functions in two different imaginary variables, $ \left( \cos(\theta) + \imath_1 \sin(\theta) \right) \left( \cos(\theta) + \imath_2 \sin(\theta) \right) $. If there are an even total number of imaginary variables, then the wrap-around pair of cis functions has a negative sin: $ \left( \cos(\theta) - \imath_1 \sin(\theta) \right) \left( \cos(\theta) + \imath_{\text{last}} \sin(\theta) \right) $. The period of these rotations is $ \pi $. Each odd prime power is assigned its own pair of cis functions. Four has one pair. Eight and greater has two pairs. The rotations are evenly split to assign a coprime residue class to each residue class of the prime.

(Proven) Hyperimaginary eigenvectors congruent to other hyperimaginary eigenvectors over multiplication have an exponential operation that will convert one to the other.

(Proven) The above Hurwitz zeta functions can be multiplied by eigenvectors to produce "hyperimaginary zeta functions". The eigenvectors are chosen so that the Hurwitz zeta function's residue class is congruent. The product representation of a hyperimaginary zeta function is $ \prod_{p \bot m}^{\mathbb{P}} \left( 1- \theta_p^x p^{-s} \right)^{-1} $

(Proven) Each hyperimaginary binomial (as a term above) has other values for theta such that $ \theta_1^x = \theta_2 $, such that when those binomials are multiplied together, a real polynomial results. $ x $ isn't necessarily complex.

(Conjectured) All of those real polynomials can be nicely represented by real binomials, like those that produce the Burgess zeta functions.

(Proven for some cases, conjectured for all) Those hyperimaginary zeta functions can be multiplied together to create Burgess zeta functions. In easy cases where there are only two residue classes, such as $ 1 \% 3 $ (positive) and $ 2 \% 3 $ (negative), this may be achieved with only positive-and-negative eigenvectors. Using the ratios of different multiplication formulae, Burgess zeta functions can be isolated completely in terms of Hurwitz zeta, Riemann zeta, and trivial terms. A few examples of the easy duplication formula to help get the point across are:

$ \left( \zeta \left( s, \frac{1}{3} \right) - \zeta \left( s, \frac{2}{3} \right) \right) 3^{-s} \zeta_{\text{p } 2 \% 3 }(s)^2 = \zeta_{\text{p } 1 \% 3 }(s) \zeta_{\text{p } 2 \% 3 }(2s) $

and

$ \left( \zeta \left( s, \frac{1}{5} \right) - \zeta \left( s, \frac{2}{5} \right) - \zeta \left( s, \frac{3}{5} \right) + \zeta \left( s, \frac{4}{5} \right) \right) 5^{-s} \zeta_{\text{p } 2,3 \% 5 }(s)^2 = \zeta_{\text{p } 1,4 \% 5 }(s) \zeta_{\text{p } 2,3 \% 5 }(2s) $

(Conjectured) It should take three hyperimaginary variables to isolate individual Burgess zeta functions by their multiplicative order modulo each prime in the modulus, more imaginary variables should allow isolation of even those roots, and four imaginary variables to isolate the Riemann zeta function in terms of Hurwitz zeta functions.

(Proven for some cases, conjectured for all) The above products of hyperimaginary zeta functions have an easy reflection formula, based on the reflection formula of the Hurwitz zeta function.

(Proven) The multiplication formulae put a stronger bound on the location of zeros in the critical strip of the Riemann zeta function, and imply a zero somewhere in the hyperimaginary zeta product function, or an obvious trail of zeros or poles from the origin to the zero that's off the critical line in Burgess zeta functions.

(Possibly proven) I've checked a few instances for the Burgess zeta function, and there does not appear to be any such trail of poles or zeros, so that should indicate that the entire zero is contained in the $ 1 \% n $ Burgess zeta function for any $ n $. But I don't trust my equations or visualizations entirely on this one yet.

I mentioned I'm an amateur. I have no idea if this is a known thing, or if parts of it are but just called something else. Not to answer a question with another question, this is really more rhetorical, but would there be a good reason why if this is a known thing, that it's virtually ignored in popular literature on the topic? If it's not a known thing, why haven't we found something so obvious before? I wouldn't exactly say that taking the log of a Dirichlet character mod q is easier to comprehend. If it's new, I'd think it would open up a new avenue of study.

So, thoughts are appreciated. Ideally I'm looking for a reason not to do this for a living from now on, probably best provided in the form of links to or explanation of existing work on the topic. I should be able to explain more if you have any questions, I'm not sure how clear all this is.