Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 1.06·2-s + 0.769·3-s + 0.375·4-s − 0.0894·5-s + 0.816·6-s − 0.431·7-s + 0.662·8-s + 0.814·9-s − 0.0948·10-s − 0.986·11-s + 0.288·12-s − 0.0213·13-s − 0.458·14-s − 0.0688·15-s + 0.546·16-s − 0.856·17-s + 0.864·18-s + 0.144·19-s − 0.0335·20-s − 0.332·21-s − 1.04·22-s + 0.543·23-s + 0.510·24-s + 0.00800·25-s − 0.0226·26-s + 1.56·27-s − 0.161·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 274625 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(274625\) = \(5^{3} \cdot 13^{3}\) |
| Sign: | $-1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 274625,\ (\ :1.5, 0.5),\ -1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1+5^{ -s})^{-1}(1+13^{ -s})^{-1}\prod_{p \nmid 65 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.