Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.192·3-s − 0.0894·5-s + 0.0370·9-s − 0.938·13-s − 0.0172·15-s − 0.171·17-s + 1.06·19-s + 0.00800·25-s + 0.00712·27-s − 0.845·29-s − 0.0926·31-s + 0.622·37-s − 0.180·39-s − 1.05·41-s − 0.993·43-s − 0.00331·45-s − 0.0329·51-s − 1.08·53-s + 0.204·57-s + 0.461·61-s + 0.0839·65-s − 0.860·67-s + 0.455·73-s + 0.00153·75-s − 1.07·79-s + 0.00137·81-s − 0.349·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 7^{4}\) |
| Sign: | $1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 7^{4} ,\ ( \ : 1.5, 0.5 ),\ 1 )\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1}(1+5^{ -s})^{-1} \prod_{p \nmid 47040 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.