Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | − 2-s + 0.333·3-s + 2·4-s + 0.800·5-s − 0.333·6-s − 0.857·7-s − 2·8-s − 0.222·9-s − 0.800·10-s − 0.181·11-s + 0.666·12-s + 0.230·13-s + 0.857·14-s + 0.266·15-s + 3·16-s − 0.470·17-s + 0.222·18-s + 0.0526·19-s + 1.60·20-s − 0.285·21-s + 0.181·22-s − 23-s − 0.666·24-s − 0.160·25-s − 0.230·26-s + 0.814·27-s − 1.71·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(361\) = \(19^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 361,\ (1:1.0),\ 1)$ |
Euler product
\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-19^{- s})^{-1}\prod_{p \nmid 19 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 0.9630442737\]
\[L(1, E, \mathrm{sym}^{2}) \approx 0.9279031293\]