Dirichlet series
| $L(s,f)$ = 1 | + 1.34·2-s − 0.577·3-s + 0.802·4-s − 0.0624·5-s − 0.775·6-s + 0.753·7-s − 0.265·8-s + 0.333·9-s − 0.0838·10-s − 0.408·11-s − 0.463·12-s + 0.527·13-s + 1.01·14-s + 0.0360·15-s − 1.15·16-s − 0.967·17-s + 0.447·18-s + 1.56·19-s − 0.0501·20-s − 0.435·21-s − 0.548·22-s + 0.285·23-s + 0.153·24-s − 0.996·25-s + 0.708·26-s − 0.192·27-s + 0.604·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+5.09i) \, \Gamma_{\R}(s-5.09i) \, L(s,f)\cr
=\mathstrut & \, \Lambda(1-s,f)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(3\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 3,\ (5.09874190873i, -5.09874190873i:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}
\end{aligned}\]