I am trying to find the functional equation for this:
$$\zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$
Therefore I let:
$$x_1=\left(1-\frac{1}{n^{s-1}}\right)$$
which I substitute with $s\rightarrow1-s$:
$$x_2=\left(1-\frac{1}{n^{(1-s-1)}}\right)$$
Then I considered:
$$\chi_{_1}(s)=\pi ^{-\frac{s}{2}} \Gamma \left(\frac{s}{2}\right) \zeta (s)$$
which I substitute with $s\rightarrow1-s$:
$$\chi_{_2}(s)=\pi ^{-\frac{1}{2} (1-s)} \Gamma \left(\frac{1-s}{2}\right) \zeta (1-s)$$
It is known that $$\chi_{_1}(s)=\chi_{_2}(s)$$ Or as the functional equation usually is stated: $$\chi(s)=\chi(1-s)$$
I then conjectured the following relationship: $$\chi_{_1}(s)x_1=-\frac{n^{-s} \left(n^s-n\right) \left(n^s-n+x_2\right)}{n-1}\chi_{_2}(s)$$
One then would have the term $$x_2\chi_{_2}(s)$$ in the numerator on the right hand side.
Can this conjectured relationship be made into a functional equation? I am new to functional equations so please bear with me if this is not entirely clearly asked.
(*Mathematica 8 start*)
n = 5
s = Log[2*Pi] + Sqrt[2]*I;
x1 = (1 - 1/n^(s - 1));
x2 = (1 - 1/n^(1 - s - 1));
chi1 = N[Pi^(-s/2)*Gamma[s/2]*Zeta[s]]
chi2 = N[Pi^(-(1 - s)/2)*Gamma[(1 - s)/2]*Zeta[(1 - s)]]
N[x1]*N[chi1]
N[-1/(-1 + n)n^-s (-n + n^s) (-n + n^s + x2 ) ]*N[chi2]
(*Mathematica 8 end*)