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  − 1.63·2-s − 1.43·3-s + 1.66·4-s − 0.0180·5-s + 2.33·6-s − 1.08·7-s − 1.08·8-s + 1.05·9-s + 0.0295·10-s − 0.917·11-s − 2.38·12-s + 0.455·13-s + 1.77·14-s + 0.0258·15-s + 0.105·16-s − 0.604·17-s − 1.71·18-s + 0.998·19-s − 0.0300·20-s + 1.56·21-s + 1.49·22-s − 0.208·23-s + 1.55·24-s − 0.999·25-s − 0.743·26-s − 0.0758·27-s − 1.81·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.39i)) \, \Gamma_{\R}(s+(1 - 1.39i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 23,\ (1 + 1.39333714148i, 1 - 1.39333714148i:\ ),\ -1)\)

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