Dirichlet series
| $L(s, E, \mathrm{sym}^{4})$ = 1 | − 0.250·2-s + 0.111·3-s − 0.312·4-s + 0.0400·5-s − 0.0277·6-s + 0.0204·7-s + 0.546·8-s + 0.0123·9-s − 0.0100·10-s + 0.00826·11-s − 0.0347·12-s + 0.171·13-s − 0.00510·14-s + 0.00444·15-s + 0.136·16-s − 0.868·17-s − 0.00308·18-s − 0.817·19-s − 0.0125·20-s + 0.00226·21-s − 0.00206·22-s + 0.395·23-s + 0.0607·24-s + 0.00160·25-s − 0.0428·26-s + 0.00137·27-s − 0.00637·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{4})\end{aligned}\]
Invariants
| Degree: | \(5\) |
| Conductor: | \(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((5,\ 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( 0 : 2.0, 1.0 ),\ 1 )\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{4}) = (1-3^{- s})^{-1}(1-5^{- s})^{-1}(1-7^{- s})^{-1}(1-11^{- s})^{-1}\prod_{p \nmid 1155 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\)
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.