Dirichlet series
| $L(s,f)$ = 1 | − 1.19·2-s + 0.577·3-s + 0.439·4-s + 1.36·5-s − 0.692·6-s + 0.135·7-s + 0.672·8-s + 0.333·9-s − 1.63·10-s + 1.04·11-s + 0.253·12-s − 0.261·13-s − 0.161·14-s + 0.788·15-s − 1.24·16-s + 0.390·17-s − 0.399·18-s + 0.813·19-s + 0.600·20-s + 0.0779·21-s − 1.25·22-s − 0.781·23-s + 0.388·24-s + 0.865·25-s + 0.313·26-s + 0.192·27-s + 0.0593·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+(1 + 7.75i)) \, \Gamma_{\R}(s+(1 - 7.75i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(3\) |
| Sign: | $1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 3,\ (1 + 7.75813319502i, 1 - 7.75813319502i:\ ),\ 1)\) |
Euler product
\(L(s,f) = \displaystyle\prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\)
Imaginary part of the first few zeros on the critical line