Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 0.431·7-s + 0.657·11-s + 0.938·13-s + 0.856·17-s + 0.0120·19-s − 1.30·23-s − 0.845·29-s − 1.06·31-s − 1.15·37-s − 0.594·41-s + 2.46·43-s − 0.180·49-s + 1.08·53-s − 0.461·61-s + 0.860·67-s − 1.04·71-s + 0.455·73-s − 0.284·77-s + 0.239·79-s − 1.07·83-s + 1.01·89-s − 0.405·91-s − 2.44·97-s + 1.00·101-s − 0.765·103-s − 0.758·107-s + 1.03·109-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 19^{3}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 19^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1-19^{- s})^{-1}\prod_{p \nmid 273600 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.