Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 4-s + 0.0539·7-s − 2·9-s − 0.411·11-s − 1.28·13-s + 16-s − 1.02·17-s + 0.144·19-s − 1.00·23-s − 0.0539·28-s + 0.941·29-s − 0.672·31-s + 2·36-s − 0.777·37-s + 1.00·41-s + 0.918·43-s + 0.411·44-s + 0.558·47-s + 0.00291·49-s + 1.28·52-s + 0.155·53-s − 2.40·59-s − 0.889·61-s − 0.107·63-s − 64-s − 0.683·67-s + 1.02·68-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(857500\) = \(2^{2} \cdot 5^{4} \cdot 7^{3}\) |
| Sign: | $-1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 857500,\ (\ :1.5, 0.5),\ -1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1+8\ 2^{-2 s})^{-1} (1-7^{- s})^{-1}\prod_{p \nmid 700 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.