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  − 1.19·3-s + 1.09·5-s − 1.74·7-s + 0.439·9-s − 1.10·11-s + 0.108·13-s − 1.30·15-s + 0.824·17-s − 0.0899·19-s + 2.09·21-s + 1.17·23-s + 0.189·25-s + 0.672·27-s − 0.590·29-s + 1.95·31-s + 1.32·33-s − 1.90·35-s + 0.198·37-s − 0.130·39-s + 0.452·41-s + 0.152·43-s + 0.479·45-s − 1.01·47-s + 2.04·49-s − 0.989·51-s − 0.635·53-s − 1.20·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 8 ^{s/2} \, \Gamma_{\R}(s+(1 + 11.4i)) \, \Gamma_{\R}(s+(1 - 11.4i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 8,\ (1 + 11.4110059735i, 1 - 11.4110059735i:\ ),\ 1)\)

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