Dirichlet series
| $L(s, E, \mathrm{sym}^{4})$ = 1 | + 2-s − 1.22·3-s + 3·4-s − 1.15·5-s − 1.22·6-s + 0.591·7-s + 3·8-s + 0.679·9-s − 1.15·10-s − 0.785·11-s − 3.66·12-s − 1.17·13-s + 0.591·14-s + 1.41·15-s + 6·16-s − 0.307·17-s + 0.679·18-s + 0.00277·19-s − 3.47·20-s − 0.723·21-s − 0.785·22-s + 23-s − 3.66·24-s + 1.11·25-s − 1.17·26-s + 0.980·27-s + 1.77·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 130321 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]
Invariants
| \( d \) | = | \(5\) |
| \( N \) | = | \(130321\) = \(19^{4}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(5,\ 130321,\ (0:2.0, 1.0),\ 1)$ |
Euler product
\[\begin{aligned}L(s, E, \mathrm{sym}^{4}) = (1-19^{- s})^{-1}\prod_{p \nmid 19 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\end{aligned}\]
Particular Values
L(1/2): not computed
L(1): not computed
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.