# Special Values of a L function

Suppose $f$ is the following modular forms

$$f=\eta(q^5)^4 [\eta(q)^4+5\,\eta(q)^3\eta(q^{25})+20 \eta(q)^2\eta(q^{25})^2+25\eta(q)\eta(q^{25})^3+25 \eta(q^{25})^4]$$

where $\eta$ is the Dedekind's eta function, i.e. $$\eta(q)=q^{\frac{1}{24}} \prod_{n=1}^{\infty}(1-q^n)$$

$f$ is of weight $4$ and level 25 (with group $\Gamma_0(25)$). I want to find whether the twisted $L$ functions, $L(f,\chi,s)$ vanishes at $s=2$ or not, where $\chi$ is the quadratic character $(./5)$. I tried to show this use modular symbols, but have not made any progress. Anyone got any ideas?

In the note, Overconvergent Modular Symbols by Robert Pollack,

http://math.bu.edu/people/rpollack/Papers/Overconvergent_modular_symbols.pdf

In page 3 there is a formula

$$L(g,\chi,1)=\frac{\tau(\chi)}{N} \sum_{a\, \text{mod}\, N} \bar{\chi}(a)2 \,\pi \,i \int_{i\,\infty}^{-a/N}g(z)dz$$

where $g$ is a modular form of weight 1. I am wondering whether there is a formula for weight 4. Then we could transform $L(f,\chi,2)$ into a sum of integrals, which might be helpful.

• If this is an eigenform, there are formulas that are valid for any weight. Look up Waldspurger's formulas for central L-values. – Kimball Jan 25 '17 at 17:36
• There is indeed an integral formula for all modular forms of any weight. Start from the identity $f_\chi = \frac{\tau(\chi)}{m} \cdot \sum_{a \mathrm{mod} m} \overline{\chi}(-a) f(z+a/m)$ (which is just Fourier transform) and integrate $f_\chi(z) z \mathrm{d}z$ from 0 to $i\infty$. Here $\chi$ is a primitive Dirichlet character mod $m$ and $f_\chi = \sum_{n \geq 1} a_n \chi(n) q^n$. – François Brunault Jan 26 '17 at 8:33
• Now for your question, the integral $\int_{a/m}^{i\infty} f(z) z \mathrm{d}z$ is an integral over the modular symbol $\{a/m,i\infty\}$. This is a period of $f$ and there are algorithms to compute them, see William Steins's PhD thesis Explicit approaches to modular abelian varieties – François Brunault Jan 26 '17 at 8:39
• @FrançoisBrunault Thank you very much for your reply, do you have a reference for this formula? – Wenzhe Feb 3 '17 at 22:29
• This is stated without proof in Manin's article Parabolic points and zeta functions of modular curves (see the first formula in the proof of Thm 3.9) wstein.org/edu/Fall2003/252/references/Manin-Parabolic/… (in this article $f$ has weight 2, but the assertion holds for any weight) This is just the Fourier decomposition of a primitive Dirichlet character, I suggest that you work it out on your own or look at math.stackexchange.com/questions/27489/… – François Brunault Feb 4 '17 at 18:12

The $L$-series vanishes at the central point (which I presume is $s = 2$ in the OP's normalization, although I'm not sure).
The reason for this is that $f \otimes \chi_{5}$ has eigenvalue equal to $-1$ under the Fricke involution $\begin{bmatrix} 0 & -1 \\ 25 & 0 \end{bmatrix}$ (according to Magma), and the sign of the functional equation of a weight $k$ newform is $i^{k}$ times the Fricke eigenvalue (by Theorem 7.1 of Iwaniec's Topics in Classical Automorphic Forms).