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  − 1.42·3-s − 0.178·5-s − 0.631·7-s + 1.04·9-s − 1.14·11-s + 0.453·13-s + 0.255·15-s + 0.390·17-s − 1.61·19-s + 0.902·21-s + 1.02·23-s − 0.968·25-s − 0.0611·27-s − 0.288·29-s − 0.330·31-s + 1.63·33-s + 0.112·35-s − 1.41·37-s − 0.647·39-s + 0.956·41-s + 0.222·43-s − 0.186·45-s + 1.08·47-s − 0.601·49-s − 0.557·51-s − 1.23·53-s + 0.205·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+5.87i) \, \Gamma_{\R}(s-5.87i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 4,\ (5.87935415776i, -5.87935415776i:\ ),\ -1)\)

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