Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 1.07·5-s − 0.0539·7-s − 2.30·11-s − 1.28·13-s + 0.856·17-s + 1.06·19-s + 0.761·23-s + 0.912·25-s + 0.307·29-s + 1.06·31-s + 0.0579·35-s + 1.01·37-s + 0.685·41-s − 0.993·43-s − 0.968·47-s + 0.00291·49-s + 0.933·53-s + 2.47·55-s + 0.688·59-s + 0.495·61-s + 1.37·65-s − 0.218·67-s + 1.06·71-s + 0.455·73-s + 0.124·77-s − 1.07·79-s − 0.918·85-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 7112448 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(7112448\) = \(2^{8} \cdot 3^{4} \cdot 7^{3}\) |
| Sign: | $-1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 7112448,\ (\ :1.5, 0.5),\ -1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1+7^{ -s})^{-1}\prod_{p \nmid 1008 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.