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  − 0.922·3-s + 0.863·5-s + 1.80·7-s − 0.149·9-s + 0.636·11-s + 0.177·13-s − 0.796·15-s + 0.461·17-s + 0.992·19-s − 1.66·21-s + 0.687·23-s − 0.254·25-s + 1.06·27-s − 0.482·29-s − 0.707·31-s − 0.587·33-s + 1.55·35-s + 0.0652·37-s − 0.163·39-s − 0.917·41-s + 0.196·43-s − 0.129·45-s − 0.390·47-s + 2.24·49-s − 0.425·51-s − 1.70·53-s + 0.550·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+(1 + 8.52i)) \, \Gamma_{\R}(s+(1 - 8.52i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 4,\ (1 + 8.52250301688i, 1 - 8.52250301688i:\ ),\ 1)\)

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