Properties

Degree $2$
Conductor $22$
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 − 1.17·3-s + 0.5·4-s − 0.627·5-s − 0.831·6-s + 1.00·7-s + 0.353·8-s + 0.382·9-s − 0.443·10-s − 0.301·11-s − 0.587·12-s + 0.472·13-s + 0.711·14-s + 0.737·15-s + 0.249·16-s + 0.612·17-s + 0.270·18-s − 1.72·19-s − 0.313·20-s − 1.18·21-s − 0.213·22-s + 0.0370·23-s − 0.415·24-s − 0.606·25-s + 0.334·26-s + 0.725·27-s + 0.502·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 22 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.04i)) \, \Gamma_{\R}(s+(1 - 1.04i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 22,\ (1 + 1.04598060261i, 1 - 1.04598060261i:\ ),\ 1)\)

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