Dirichlet series
| $L(s, E, \mathrm{sym}^{4})$ = 1 | − 0.250·2-s + 0.111·3-s − 0.312·4-s − 0.760·5-s − 0.0277·6-s − 0.632·7-s + 0.546·8-s + 0.0123·9-s + 0.190·10-s + 0.00826·11-s − 0.0347·12-s + 0.171·13-s + 0.158·14-s − 0.0844·15-s + 0.136·16-s + 0.349·17-s − 0.00308·18-s + 19-s + 0.237·20-s − 0.0702·21-s − 0.00206·22-s + 0.395·23-s + 0.0607·24-s − 0.334·25-s − 0.0428·26-s + 0.00137·27-s + 0.197·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]
Invariants
| Degree: | \(5\) |
| Conductor: | \(1185921\) = \(3^{4} \cdot 11^{4}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((5,\ 1185921,\ (0:2.0, 1.0),\ 1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{4}) = (1-3^{- s})^{-1}(1-11^{- s})^{-1}\prod_{p \nmid 33 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\)
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.