Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.192·3-s − 1.07·5-s − 0.0539·7-s + 0.0370·9-s + 0.938·13-s − 0.206·15-s + 0.171·17-s + 1.06·19-s − 0.0103·21-s + 1.08·23-s + 0.912·25-s + 0.00712·27-s − 0.845·29-s − 0.0926·31-s + 0.0579·35-s − 1.15·37-s + 0.180·39-s − 0.685·41-s + 2.46·43-s − 0.0397·45-s + 0.744·47-s + 0.00291·49-s + 0.0329·51-s − 1.08·53-s + 0.204·57-s − 0.900·59-s + 0.461·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 592704 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(592704\) = \(2^{6} \cdot 3^{3} \cdot 7^{3}\) |
| Sign: | $-1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 592704,\ (\ :1.5, 0.5),\ -1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1}(1+7^{ -s})^{-1}\prod_{p \nmid 168 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.