How to find minimum length of a FIR symmetric filter if I am given DTFT

I am practising for upcoming exams and came across this question.

Let $$h[n]$$ be an FIR filter such that $$h[n] = 0$$ when $$|n| > M$$ and $$h[n] = h[−n]$$. A plot of $$H(e^{j\omega})$$ (DTFT of $$h[n]$$) is shown below. The eventual goal is to implement this filter digitally. Then, with the given information, is it possible to determine the minimum value of $$M$$? Yes or no, with a suitable technical explanation.

What I though was that if I knew the order of this filter I could have found the length of the FIR filter as well. However order cannot be extracted from this graph.

HINT: How many zeros of $$H(e^{j\omega})$$ do you count? Does that say anything about the (minimum) order of the corresponding polynomial?
• @PulkitJoshi: I think there's a double zero at $\omega=0$. And yes you're right, you can only see the zeros on the unit circle, but note that due to the symmetry of $h[n]$ you can only have zeros either on the unit circle or mirrored at the unit circle (the latter you wouldn't see directly in the frequency response plot). – Matt L. Jul 31 '20 at 15:36
• @PulkitJoshi: If the curve touches zero at $\omega=0$ then it's a double zero. – Matt L. Jul 31 '20 at 16:19