# Nyquist plot of a time delay open-loop transfer function

As shown in the example 6 of here, the Nyquist plot of a open-loop transfer function that has a delay term in the numerator will encircle the origin several times (in fact, infinitely many times). However, is that a contradiction to the argument principle?

To be specific, that example is repeated below:

$$\L(s)=90e^{-0.05s}/(s+3)(s+6)\$$

Since $$\L(s)\$$ has no poles neither zeros in the RHP, and since $$\e^{-0.05s}\$$ is an entire function, according to the argument principle, the number of encirclement of the origin equals $$\N-P=0\$$, where N and P are the numbers of zeros and poles of $$\L(s)\$$ in the RHP.