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 − 0.252·3-s + 0.5·4-s + 0.306·5-s + 0.178·6-s + 1.57·7-s − 0.353·8-s − 0.936·9-s − 0.216·10-s − 0.301·11-s − 0.126·12-s + 0.463·13-s − 1.11·14-s − 0.0772·15-s + 0.250·16-s − 0.759·17-s + 0.662·18-s + 1.62·19-s + 0.153·20-s − 0.397·21-s + 0.213·22-s + 0.419·23-s + 0.0891·24-s − 0.906·25-s − 0.327·26-s + 0.488·27-s + 0.787·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 22 ^{s/2} \, \Gamma_{\R}(s+1.72i) \, \Gamma_{\R}(s-1.72i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 22,\ (1.72969651767i, -1.72969651767i:\ ),\ 1)\)

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