Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.0894·5-s − 0.0539·7-s + 0.657·11-s − 0.938·13-s + 0.856·17-s + 1.06·19-s − 1.30·23-s + 0.00800·25-s − 0.845·29-s + 0.0926·31-s − 0.00482·35-s − 0.622·37-s + 0.594·41-s − 2.46·43-s + 0.744·47-s + 0.00291·49-s − 1.08·53-s + 0.0588·55-s + 0.900·59-s − 0.495·61-s − 0.0839·65-s + 0.218·67-s − 1.04·71-s − 1.12·73-s − 0.0355·77-s + 0.349·83-s + 0.0765·85-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2^{12} \cdot 3^{4} \cdot 5^{3} \cdot 7^{3}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 2^{12} \cdot 3^{4} \cdot 5^{3} \cdot 7^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1-5^{- s})^{-1}(1+7^{ -s})^{-1}\prod_{p \nmid 20160 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.