Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | − 0.5·2-s + 0.333·3-s + 0.750·4-s + 0.200·5-s − 0.166·6-s + 1.28·7-s + 0.375·8-s − 0.222·9-s − 0.100·10-s − 0.636·11-s + 0.250·12-s + 0.0769·13-s − 0.642·14-s + 0.0666·15-s − 0.312·16-s − 0.764·17-s + 0.111·18-s + 0.894·19-s + 0.149·20-s + 0.428·21-s + 0.318·22-s + 0.565·23-s + 0.125·24-s + 0.0400·25-s − 0.0384·26-s + 0.814·27-s + 0.964·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(4225\) = \(5^{2} \cdot 13^{2}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((3,\ 4225,\ (1:1.0),\ 1)\) |
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.125409076\]
\[L(1, E, \mathrm{sym}^{2}) \approx 1.322954445\]
Euler product
\(L(s, E, \mathrm{sym}^{2}) = (1-5^{- s})^{-1}(1-13^{- s})^{-1}\prod_{p \nmid 65 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)
Imaginary part of the first few zeros on the critical line