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  − 1.79·2-s + 0.465·3-s + 2.23·4-s − 0.447·5-s − 0.837·6-s + 1.15·7-s − 2.22·8-s − 0.783·9-s + 0.804·10-s − 0.767·11-s + 1.04·12-s − 0.0126·13-s − 2.08·14-s − 0.208·15-s + 1.77·16-s − 0.371·17-s + 1.41·18-s − 1.56·19-s − 1.00·20-s + 0.537·21-s + 1.38·22-s − 0.425·23-s − 1.03·24-s + 0.199·25-s + 0.0228·26-s − 0.829·27-s + 2.58·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s+(1 + 8.22i)) \, \Gamma_{\R}(s+(1 - 8.22i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 5,\ (1 + 8.2261459726i, 1 - 8.2261459726i:\ ),\ -1)\)

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