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  − 0.951·2-s − 0.0317·3-s − 0.0946·4-s + 1.04·5-s + 0.0301·6-s + 0.607·7-s + 1.04·8-s − 0.998·9-s − 0.990·10-s + 0.513·11-s + 0.00299·12-s − 1.14·13-s − 0.577·14-s − 0.0329·15-s − 0.896·16-s + 1.55·17-s + 0.950·18-s + 0.705·19-s − 0.0984·20-s − 0.0192·21-s − 0.488·22-s − 0.208·23-s − 0.0330·24-s + 0.0829·25-s + 1.08·26-s + 0.0633·27-s − 0.0574·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1.57i) \, \Gamma_{\R}(s-1.57i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 23,\ (1.57958924015i, -1.57958924015i:\ ),\ 1)\)

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