Dirichlet series
| $L(s,f)$ = 1 | + 0.888·2-s − 1.51·3-s − 0.211·4-s + 0.374·5-s − 1.34·6-s − 0.686·7-s − 1.07·8-s + 1.28·9-s + 0.332·10-s + 0.301·11-s + 0.319·12-s − 0.897·13-s − 0.610·14-s − 0.566·15-s − 0.744·16-s + 1.66·17-s + 1.14·18-s − 1.91·19-s − 0.0792·20-s + 1.03·21-s + 0.267·22-s − 0.339·23-s + 1.62·24-s − 0.859·25-s − 0.796·26-s − 0.433·27-s + 0.145·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+4.55i) \, \Gamma_{\R}(s-4.55i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(11\) |
| Sign: | $-1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 11,\ (4.55526078532i, -4.55526078532i:\ ),\ -1)\) |
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