Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | + 0.5·2-s + 0.333·3-s + 0.250·4-s − 5-s + 0.166·6-s + 1.28·7-s + 0.125·8-s − 0.222·9-s − 0.5·10-s + 2.27·11-s + 0.0833·12-s − 0.692·13-s + 0.642·14-s − 0.333·15-s + 0.0625·16-s + 0.0588·17-s − 0.111·18-s − 0.157·19-s − 0.250·20-s + 0.428·21-s + 1.13·22-s − 23-s + 0.0416·24-s + 2·25-s − 0.346·26-s + 0.814·27-s + 0.321·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1156\) = \(2^{2} \cdot 17^{2}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((3,\ 1156,\ (1:1.0),\ 1)\) |
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.027457844\]
\[L(1, E, \mathrm{sym}^{2}) \approx 1.548750374\]
Euler product
\(L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-9\ 7^{- s}+63\ 7^{-2 s}-343 \ 7^{-3 s})^{-1}(1-17^{- s})^{-1}\prod_{p \nmid 53312 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)
Imaginary part of the first few zeros on the critical line