Properties

Degree $2$
Conductor $7$
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 − 0.838·3-s + 2.23·4-s − 0.251·5-s − 1.50·6-s − 0.377·7-s + 2.22·8-s − 0.297·9-s − 0.452·10-s + 0.929·11-s − 1.87·12-s + 0.444·13-s − 0.680·14-s + 0.210·15-s + 1.76·16-s + 0.0943·17-s − 0.535·18-s − 0.688·19-s − 0.562·20-s + 0.316·21-s + 1.67·22-s + 0.0987·23-s − 1.86·24-s − 0.936·25-s + 0.799·26-s + 1.08·27-s − 0.845·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s+4.02i) \, \Gamma_{\R}(s-4.02i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 7,\ (4.02170481942i, -4.02170481942i:\ ),\ 1)\)

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