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 + 0.633·5-s + 1.52·7-s + 0.333·9-s + 0.381·11-s − 1.13·13-s + 0.365·15-s + 1.71·17-s − 1.02·19-s + 0.880·21-s − 1.27·23-s − 0.598·25-s + 0.192·27-s + 0.905·29-s − 0.247·31-s + 0.220·33-s + 0.966·35-s − 0.182·37-s − 0.654·39-s − 0.0773·41-s − 1.21·43-s + 0.211·45-s − 0.932·47-s + 1.32·49-s + 0.991·51-s + 0.339·53-s + 0.242·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 12 ^{s/2} \, \Gamma_{\R}(s+4.93i) \, \Gamma_{\R}(s-4.93i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 12,\ (4.9342604018i, -4.9342604018i:\ ),\ 1)\)

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