Properties

Degree $2$
Conductor $8$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.710·3-s − 1.57·5-s + 0.855·7-s − 0.494·9-s − 0.289·11-s − 0.511·13-s + 1.12·15-s + 0.163·17-s − 0.288·19-s − 0.608·21-s − 1.13·23-s + 1.49·25-s + 1.06·27-s − 1.34·29-s + 0.795·31-s + 0.205·33-s − 1.35·35-s − 0.690·37-s + 0.363·39-s + 1.03·41-s + 0.243·43-s + 0.781·45-s − 0.485·47-s − 0.267·49-s − 0.115·51-s − 1.61·53-s + 0.457·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 8 ^{s/2} \, \Gamma_{\R}(s+4.64i) \, \Gamma_{\R}(s-4.64i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 8,\ (4.64659164296i, -4.64659164296i:\ ),\ -1)\)

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