Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 0.0894·5-s − 0.0539·7-s − 2·9-s + 1.28·13-s + 0.856·17-s + 0.00800·25-s + 0.845·29-s − 0.0926·31-s + 0.00482·35-s − 1.15·37-s + 0.594·41-s − 0.993·43-s + 0.178·45-s + 0.744·47-s + 0.00291·49-s + 0.528·53-s − 0.953·59-s + 2.17·61-s + 0.107·63-s − 0.114·65-s + 0.218·67-s + 3.04·71-s + 0.455·73-s + 1.07·79-s + 3·81-s − 1.07·83-s − 0.0765·85-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{8} \cdot 5^{3} \cdot 7^{3} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2^{8} \cdot 5^{3} \cdot 7^{3} \cdot 11^{4}\) |
| Sign: | $-1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 2^{8} \cdot 5^{3} \cdot 7^{3} \cdot 11^{4} ,\ ( \ : 1.5, 0.5 ),\ -1 )\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1+5^{ -s})^{-1}(1+7^{ -s})^{-1} \prod_{p \nmid 67760 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
Imaginary part of the first few zeros on the critical line
Zeros not available.
Graph of the $Z$-function along the critical line
Plot not available.