Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.353·2-s + 0.769·3-s + 0.125·4-s + 0.272·6-s − 0.431·7-s + 0.0441·8-s + 0.814·9-s + 2.30·11-s + 0.0962·12-s − 0.938·13-s − 0.152·14-s + 0.0156·16-s − 0.0142·17-s + 0.288·18-s + 1.06·19-s − 0.332·21-s + 0.814·22-s + 0.0340·24-s − 2·25-s − 0.331·26-s + 1.56·27-s − 0.0539·28-s + 1.06·31-s + 0.00552·32-s + 1.77·33-s − 0.00504·34-s + 0.101·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 39304 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(39304\) = \(2^{3} \cdot 17^{3}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 39304,\ (\ :1.5, 0.5),\ 1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1+17^{ -s})^{-1}\prod_{p \nmid 34 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.