Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 0.353·2-s + 0.192·3-s + 0.125·4-s + 0.0894·5-s − 0.0680·6-s − 0.431·7-s − 0.0441·8-s + 0.0370·9-s − 0.0316·10-s + 0.0240·12-s − 0.938·13-s + 0.152·14-s + 0.0172·15-s + 0.0156·16-s − 0.856·17-s − 0.0130·18-s + 0.0120·19-s + 0.0111·20-s − 0.0831·21-s − 1.30·23-s − 0.00850·24-s + 0.00800·25-s + 0.331·26-s + 0.00712·27-s − 0.0539·28-s + 0.845·29-s − 0.00608·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{4} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{4} \cdot 19^{3}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{4} \cdot 19^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1}(1-3^{- s})^{-1}(1-5^{- s})^{-1} (1-19^{- s})^{-1}\prod_{p \nmid 68970 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.