Properties

Degree $2$
Conductor $12$
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 + 1.85·5-s + 0.184·7-s + 0.333·9-s + 0.459·11-s + 1.33·13-s − 1.06·15-s − 0.527·17-s + 0.490·19-s − 0.106·21-s − 0.893·23-s + 2.43·25-s − 0.192·27-s − 0.333·29-s − 1.49·31-s − 0.265·33-s + 0.341·35-s + 0.602·37-s − 0.771·39-s + 0.772·41-s + 0.797·43-s + 0.617·45-s − 0.443·47-s − 0.966·49-s + 0.304·51-s − 1.20·53-s + 0.851·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 12 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.98i)) \, \Gamma_{\R}(s+(1 - 4.98i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 12,\ (1 + 4.98315181103i, 1 - 4.98315181103i:\ ),\ 1)\)

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