Maximizing a function for $k \in [-\pi, \pi]$

Question

I want to find the maximum of $|g(\kappa)|$ defined as follows

For some $A>0$, $\alpha\in (0,\infty)$, and $\kappa \in [-\kappa^{*},\kappa^{*}]\subset [-\pi,\pi]$, define $g(\kappa )$ \begin{align} g(\kappa ) &= -\sqrt[]{\Big | \lambda - 4A^{2}\zeta(1+\alpha) -2(1+A^{2})\sum\limits_{m\in\mathbb{Z}_{>0}}\frac{\cos(\kappa m)}{m^{1+\alpha}} \Big |\Big ( \lambda - 2(1+A^{2})\sum\limits_{m\in\mathbb{Z}_{>0}}\frac{\cos(\kappa m)}{m^{1+\alpha}}\Big )}, \end{align}

Let u denote the terms within the absolute value signs. Then $|g(k)| = (|u|^{2})^{\frac{1}{2}}(\lambda - 2(1+A^{2})\sum_{m>0}\frac{\cos(km)}{m^{1+\alpha}})$. Where $\lambda = 2(1+A^{2})\zeta(1+\alpha)$.

Then $\partial_{k}|g(k)| = (\frac{1}{|u|} + |u|) (2A^{2}\sum\limits_{m>0}\frac{\sin(km)}{m^{1+\alpha}})$

This vanishes with either or both terms are zero. Immediately I see $k = \pm \pi$. However, for $A = 2, \alpha = 3$ I plot $|g(k)|$ and see that is not the case. Why?

Clear["Global`*"]

f[k_, \[Alpha]_] := Sum[Cos[k*m]/m^(1 + \[Alpha]), {m, 1, Infinity}]
Plot[(Abs[
     2 (1 + A^2) Zeta[1 + \[Alpha]] - 4 A^2 Zeta[1 + \[Alpha]] - 
      2 (1 + A^2) f[k, \[Alpha]]]) (2 (1 + A^2) Zeta[1 + \[Alpha]] - 
     2 (1 + A^2) f[k, \[Alpha]]) /. {A -> 2, \[Alpha] -> 
    3}, {k, -\[Pi], \[Pi]}]