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 + 0.175·5-s + 0.729·7-s + 0.333·9-s − 1.34·11-s − 0.168·13-s − 0.101·15-s + 1.32·17-s + 0.209·19-s − 0.421·21-s + 0.541·23-s − 0.969·25-s − 0.192·27-s − 1.13·29-s + 0.543·31-s + 0.779·33-s + 0.128·35-s + 1.26·37-s + 0.0972·39-s − 0.616·41-s − 0.843·43-s + 0.0585·45-s − 0.234·47-s − 0.467·49-s − 0.762·51-s + 1.13·53-s − 0.236·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 12 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.69i)) \, \Gamma_{\R}(s+(1 - 1.69i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 12,\ (1 + 1.69106559844i, 1 - 1.69106559844i:\ ),\ 1)\)

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