Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 0.353·2-s − 0.962·3-s + 0.125·4-s + 0.340·6-s + 0.701·7-s − 0.0441·8-s − 0.185·9-s − 2.30·11-s − 0.120·12-s − 0.106·13-s − 0.248·14-s + 0.0156·16-s − 1.07·17-s + 0.0654·18-s + 0.0120·19-s − 0.675·21-s + 0.814·22-s − 1.00·23-s + 0.0425·24-s − 2·25-s + 0.0377·26-s + 0.285·27-s + 0.0877·28-s + 1.32·29-s + 1.06·31-s − 0.00552·32-s + 2.21·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 54872 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(54872\) = \(2^{3} \cdot 19^{3}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 54872,\ (\ :1.5, 0.5),\ 1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1}(1-19^{- s})^{-1}\prod_{p \nmid 38 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.