Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.192·3-s + 0.0894·5-s + 0.0370·9-s + 0.657·11-s + 1.28·13-s + 0.0172·15-s − 0.171·17-s + 1.06·19-s + 0.00800·25-s + 0.00712·27-s + 0.691·29-s − 0.0926·31-s + 0.126·33-s + 0.622·37-s + 0.246·39-s + 1.05·41-s + 2.46·43-s + 0.00331·45-s − 0.744·47-s − 2·49-s − 0.0329·51-s − 1.08·53-s + 0.0588·55-s + 0.204·57-s + 0.688·59-s + 2.17·61-s + 0.114·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(216000\) = \(2^{6} \cdot 3^{3} \cdot 5^{3}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 216000,\ (\ :1.5, 0.5),\ 1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1}(1-5^{- s})^{-1}\prod_{p \nmid 120 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.