Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.192·3-s + 0.804·5-s + 0.809·7-s + 0.0370·9-s − 0.575·11-s + 0.938·13-s + 0.154·15-s − 16-s − 0.856·17-s − 0.144·19-s + 0.155·21-s − 1.00·23-s − 0.792·25-s + 0.00712·27-s − 0.941·29-s − 0.672·31-s − 0.110·33-s + 0.651·35-s + 0.777·37-s + 0.180·39-s + 0.685·41-s − 0.496·43-s + 0.0298·45-s − 0.00310·47-s − 0.192·48-s + 0.860·49-s − 0.164·51-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 2803221 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2803221\) = \(3^{3} \cdot 47^{3}\) |
| Sign: | $1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 2803221,\ (\ :1.5, 0.5),\ 1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1}(1+47^{ -s})^{-1}\prod_{p \nmid 141 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.