Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | + 2-s + 0.333·3-s + 0.800·5-s + 0.333·6-s + 2.57·7-s − 0.222·9-s + 0.800·10-s + 0.454·11-s − 0.307·13-s + 2.57·14-s + 0.266·15-s + 16-s + 1.11·17-s − 0.222·18-s + 0.315·19-s + 0.857·21-s + 0.454·22-s − 0.304·23-s − 0.160·25-s − 0.307·26-s + 0.814·27-s + 0.241·29-s + 0.266·30-s − 0.483·31-s + 32-s + 0.151·33-s + 1.11·34-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 151321 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr
=\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2})
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(151321\) = \(389^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 151321,\ (1:1.0),\ 1)$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{2}) = (1-389^{- s})^{-1}\prod_{p \nmid 389 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
Particular Values
L(1/2): not computed
L(1): not computed
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.