Simultaneous deficiency and ramification for an exponential polynomial ($f=e^z(e^z-1{)}^2,{\delta }_f(0)={\theta }_f(0)=1/3$)

The rational composition law for the Nevanlinna characteristic gives $T_f(r)=3r/\pi +O(1)$. Its zeros are precisely $2\pi i\mathbb{Z}$, all of multiplicity two, so $N(r;0)=2r/\pi +O(\log r)$ and $\overline{N}(r;0)=r/\pi +O(\log r)$. Hence both its Nevanlinna deficiency and Nevanlinna ramification index at zero equal one third.