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  + 0.882·2-s − 1.52·3-s − 0.221·4-s − 0.447·5-s − 1.34·6-s + 0.812·7-s − 1.07·8-s + 1.33·9-s − 0.394·10-s − 0.787·11-s + 0.338·12-s − 0.643·13-s + 0.716·14-s + 0.683·15-s − 0.729·16-s − 0.272·17-s + 1.18·18-s − 0.770·19-s + 0.0988·20-s − 1.24·21-s − 0.694·22-s + 1.30·23-s + 1.64·24-s + 0.199·25-s − 0.567·26-s − 0.517·27-s − 0.179·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s+(1 + 6.35i)) \, \Gamma_{\R}(s+(1 - 6.35i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 5,\ (1 + 6.35122321676i, 1 - 6.35122321676i:\ ),\ -1)\)

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