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  − 1.33·2-s + 0.794·3-s + 0.788·4-s − 0.265·5-s − 1.06·6-s + 0.377·7-s + 0.283·8-s − 0.369·9-s + 0.355·10-s − 0.384·11-s + 0.625·12-s + 0.811·13-s − 0.505·14-s − 0.210·15-s − 1.16·16-s − 0.421·17-s + 0.494·18-s + 0.545·19-s − 0.209·20-s + 0.300·21-s + 0.514·22-s + 1.17·23-s + 0.224·24-s − 0.929·25-s − 1.08·26-s − 1.08·27-s + 0.297·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.92i)) \, \Gamma_{\R}(s+(1 - 1.92i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 7,\ (1 + 1.92464430511i, 1 - 1.92464430511i:\ ),\ 1)\)

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