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  + 1.52·2-s − 1.03·3-s + 1.33·4-s + 0.447·5-s − 1.58·6-s + 0.325·7-s + 0.515·8-s + 0.0803·9-s + 0.683·10-s − 0.505·11-s − 1.38·12-s + 0.904·13-s + 0.497·14-s − 0.464·15-s − 0.549·16-s + 1.06·17-s + 0.122·18-s − 0.362·19-s + 0.598·20-s − 0.338·21-s − 0.772·22-s − 0.478·23-s − 0.535·24-s + 0.199·25-s + 1.38·26-s + 0.955·27-s + 0.435·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.10i)) \, \Gamma_{\R}(s+(1 - 4.10i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 5,\ (1 + 4.10322180991i, 1 - 4.10322180991i:\ ),\ 1)\)

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