Dirichlet series
| $L(s, E, \mathrm{sym}^{4})$ = 1 | + 0.111·3-s + 0.0400·5-s + 7-s + 0.0123·9-s − 1.24·11-s + 0.360·13-s + 0.00444·15-s − 0.868·17-s − 0.817·19-s + 0.111·21-s + 23-s + 0.00160·25-s + 0.00137·27-s + 0.605·29-s − 0.931·31-s − 0.138·33-s + 0.0400·35-s + 0.687·37-s + 0.0401·39-s − 0.863·41-s + 2.16·43-s + 0.000493·45-s − 1.23·47-s + 3·49-s − 0.0965·51-s − 0.576·53-s − 0.0499·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]
Invariants
| Degree: | \(5\) |
| Conductor: | \(3240000\) = \(2^{6} \cdot 3^{4} \cdot 5^{4}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((5,\ 3240000,\ (0:2.0, 1.0),\ 1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{4}) = (1-3^{- s})^{-1}(1-5^{- s})^{-1}\prod_{p \nmid 120 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\)
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.