Properties

Degree $2$
Conductor $18$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.707·2-s + 0.5·4-s + 0.963·5-s + 0.173·7-s − 0.353·8-s − 0.681·10-s − 0.907·11-s + 0.767·13-s − 0.122·14-s + 0.250·16-s + 1.33·17-s − 1.14·19-s + 0.481·20-s + 0.642·22-s − 0.521·23-s − 0.0712·25-s − 0.542·26-s + 0.0868·28-s − 0.771·29-s + 0.736·31-s − 0.176·32-s − 0.942·34-s + 0.167·35-s − 0.307·37-s + 0.807·38-s − 0.340·40-s + 1.52·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 18 ^{s/2} \, \Gamma_{\R}(s+1.96i) \, \Gamma_{\R}(s-1.96i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 18,\ (1.9678447101i, -1.9678447101i:\ ),\ 1)\)

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