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  − 1.37·3-s + 0.214·5-s − 1.43·7-s + 0.892·9-s + 0.0785·11-s − 0.0193·13-s − 0.294·15-s − 0.944·17-s + 0.479·19-s + 1.96·21-s − 0.930·23-s − 0.954·25-s + 0.147·27-s − 0.656·29-s + 0.964·31-s − 0.108·33-s − 0.306·35-s + 1.73·37-s + 0.0265·39-s − 1.47·41-s − 1.10·43-s + 0.191·45-s − 0.144·47-s + 1.04·49-s + 1.29·51-s − 0.639·53-s + 0.0168·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 8 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.36i)) \, \Gamma_{\R}(s+(1 - 4.36i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 8,\ (1 + 4.36801954023i, 1 - 4.36801954023i:\ ),\ -1)\)

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

Zeros not available.

Graph of the $Z$-function along the critical line

Plot not available.