Let $f(z)=\sum\_{n\ge 1}a\_nq^n$ be a cusp form, where $q=e^{2\pi i z}$. Let $
L(s) = \sum\_{n\ge 1} a\_nn^{-s}
$ be its corresponding L-function. The completed L-function of $L(s)$, $\Lambda(s)$, should satisfy a functional equation $\Lambda(s)=\bar \Lambda(a-s)$ for some integer $a$.


The analytic rank of $f(s)$ is defined as the order of vanishing of $L$ at $a/2$.


**Question:** If $f$ is of odd weight, is its analytic rank $0$ (or equivalently, $L(a/2)\neq0$)?



Do odd-weight cusp forms have analytic rank 0?
It is conjectured that $L$-functions of motives do not have zeros or poles on the real line, except possibly at integers. In particular, if $f$ is a newform of odd weight $k$, then $L(f,s)$ should not vanish at $s=k/2$.


This conjecture appears in Fontaine, Perrin-Riou, *Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L*. Motives (Seattle, WA, 1991), 599–706, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. See Remark 4.5.3.i).


I don't know if this is proven for $L$-functions of modular forms.


