Dirichlet series
| $L(s, E, \mathrm{sym}^{4})$ = 1 | + 0.250·2-s + 3-s + 0.0625·4-s − 1.15·5-s + 0.250·6-s − 0.632·7-s + 0.0156·8-s + 9-s − 0.289·10-s + 11-s + 0.0625·12-s + 0.775·13-s − 0.158·14-s − 1.15·15-s + 0.00390·16-s − 0.307·17-s + 0.250·18-s − 0.817·19-s − 0.0724·20-s − 0.632·21-s + 0.250·22-s + 23-s + 0.0156·24-s + 1.11·25-s + 0.193·26-s + 27-s − 0.0395·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 104976 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]
Invariants
| Degree: | \(5\) |
| Conductor: | \(104976\) = \(2^{4} \cdot 3^{8}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((5,\ 104976,\ (0:2.0, 1.0),\ 1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{4}) = (1-2^{- s})^{-1}(1-9\ 3^{- s})^{-1}\prod_{p \nmid 162 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\)
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.