Existence of the saddle point $\tau_0$ for $r=3$, $q=13$ with the analytic hypotheses of Lemma 2

Lemma 2 applied to the parameter set of the note. For the parameters $r = 3$, $q = 13$, $\eta_0 = 91$, $\eta_1 = \eta_2 = \eta_3 = 27$, $\eta_4 = 29$, $\dots$, $\eta_{13} = 38$ of Zudilin's note, the saddle-point equation

has a root $\tau_0$ in the upper half-plane $\operatorname{Im}\tau_0 > 0$, of maximal real part among the roots in the upper half-plane, with $\operatorname{Re}\tau_0 < \eta_0$, and such that $\operatorname{Im} f_0(\tau_0) \notin \pi\mathbb{Z}$, where $f_0$ is the auxiliary function of the note. These are exactly the hypotheses under which Lemma 2 of the note yields the asymptotic rate $\limsup_{n\to\infty} \log|F_n|/n = \operatorname{Re} f_0(\tau_0) = -C_0$, and under which the leading asymptotic coefficient of $|F_n|$ does not vanish ($F_n \neq 0$ for all large $n$). The companion node ZudilinZeta.zudilin_numeric_C0_gt_C1 records the computed value $C_0 = 227.58019641\ldots$ for this root. The statement is the analytic-existence content of Lemma 2 specialised to the tuple (3, 13); it is faithful to the note and its follow-up paper (Zudilin, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002) 489–542), where the analogous saddle point is exhibited.