Properties

Degree $2$
Conductor $9$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 1.25·2-s + 0.577·4-s − 0.740·5-s − 0.853·7-s + 0.530·8-s + 0.929·10-s − 0.905·11-s + 0.0466·13-s + 1.07·14-s − 1.24·16-s − 0.633·17-s + 0.508·19-s − 0.427·20-s + 1.13·22-s + 1.02·23-s − 0.452·25-s − 0.0586·26-s − 0.492·28-s − 1.40·29-s − 1.04·31-s + 1.03·32-s + 0.795·34-s + 0.631·35-s + 1.30·37-s − 0.639·38-s − 0.392·40-s − 0.365·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 9 ^{s/2} \, \Gamma_{\R}(s+(1 + 3.53i)) \, \Gamma_{\R}(s+(1 - 3.53i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 9,\ (1 + 3.53600209294i, 1 - 3.53600209294i:\ ),\ -1)\)

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

Graph of the $Z$-function along the critical line