Properties

Degree $2$
Conductor $4$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 1.47·3-s + 0.266·5-s − 1.90·7-s + 1.18·9-s + 1.37·11-s − 1.05·13-s + 0.394·15-s + 0.635·17-s − 0.0721·19-s − 2.81·21-s − 0.558·23-s − 0.928·25-s + 0.276·27-s − 1.08·29-s − 0.836·31-s + 2.03·33-s − 0.506·35-s + 0.531·37-s − 1.56·39-s + 0.0822·41-s − 0.445·43-s + 0.280·45-s − 0.588·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+26.4i) \, \Gamma_{\R}(s-26.4i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 4,\ (26.4600305542i, -26.4600305542i:\ ),\ -1)\)

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.