Properties

Degree $2$
Conductor $23$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.668·2-s − 0.143·3-s − 0.553·4-s − 1.15·5-s + 0.0958·6-s + 1.20·7-s + 1.03·8-s − 0.979·9-s + 0.769·10-s − 1.32·11-s + 0.0794·12-s + 0.154·13-s − 0.802·14-s + 0.165·15-s − 0.139·16-s − 1.07·17-s + 0.654·18-s + 0.124·19-s + 0.637·20-s − 0.172·21-s + 0.884·22-s − 0.208·23-s − 0.148·24-s + 0.327·25-s − 0.103·26-s + 0.283·27-s − 0.665·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+(1 + 2.49i)) \, \Gamma_{\R}(s+(1 - 2.49i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 23,\ (1 + 2.49362378677i, 1 - 2.49362378677i:\ ),\ -1)\)

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