Properties

Degree $2$
Conductor $5$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.785·2-s − 0.227·3-s − 0.383·4-s − 0.447·5-s + 0.178·6-s − 0.239·7-s + 1.08·8-s − 0.948·9-s + 0.351·10-s − 1.14·11-s + 0.0873·12-s + 1.34·13-s + 0.188·14-s + 0.101·15-s − 0.469·16-s − 1.46·17-s + 0.744·18-s − 0.0589·19-s + 0.171·20-s + 0.0546·21-s + 0.898·22-s − 0.109·23-s − 0.247·24-s + 0.199·25-s − 1.05·26-s + 0.443·27-s + 0.0919·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.89i)) \, \Gamma_{\R}(s+(1 - 4.89i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 5,\ (1 + 4.89723501573i, 1 - 4.89723501573i:\ ),\ -1)\)

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