Properties

Degree $2$
Conductor $24$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 0.577·3-s − 0.508·5-s − 0.971·7-s + 0.333·9-s + 0.347·11-s + 1.59·13-s − 0.293·15-s − 0.671·17-s + 0.346·19-s − 0.560·21-s − 0.474·23-s − 0.741·25-s + 0.192·27-s − 0.0103·29-s + 0.482·31-s + 0.200·33-s + 0.493·35-s − 0.420·37-s + 0.920·39-s + 1.99·41-s − 1.13·43-s − 0.169·45-s − 1.46·47-s − 0.0570·49-s − 0.387·51-s − 0.857·53-s − 0.176·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.30i)) \, \Gamma_{\R}(s+(1 - 1.30i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 24,\ (1 + 1.30086635839i, 1 - 1.30086635839i:\ ),\ 1)\)

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