Doubts about Notch filter design I am following the process that is described in this question: Transfer function of second order notch filter , I want to create a notch filter with the band suppressed equal to $f_c = 4000$ Hz, so using $\omega_n= f_c / f_s $, ($f_s = 48000$), I obtained the $\omega_n = \frac{\pi}{6}$, then using the exact same formula, with $a =0.8$. The pole-zero graph I get has the zero in $1$ and a pole in $0.8$, is this correct?? I am getting the half of the filter since the filter is centered in $0$ and not in $4000$ Hz.

As far as I know it must be centered in the wn I have (based on $f =4000$ Hz), but I am not sure why it is centered in $0$, or how to center it in the desired frequency.
I get a pole zero graph like this one, with zeros in $1$ and poles in $0.8$.



Please advice
This Octave / Matlab code gives you a 2nd ord notch filter at $\omega_n = \pi/6$

r = 0.99;          % notch radius (closer to 1 stiffer)
wn = pi/6;         % notch radian frequency...  

% Create the 2nd order NOTCH filter coefficients b() and a()
b = r*conv([1,  -exp(j*wn)],[1,  -exp(-j*wn)]);      
a = conv([1,  -r*exp(j*wn)],[1,  -r*exp(-j*wn)]);    

figure,freqz(b,a,2048);


with the following result: