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  − 1.02·2-s − 0.0177·3-s + 0.0427·4-s − 0.723·5-s + 0.0180·6-s − 0.377·7-s + 0.977·8-s − 0.999·9-s + 0.738·10-s + 0.416·11-s − 0.000758·12-s − 1.41·13-s + 0.385·14-s + 0.0128·15-s − 1.04·16-s + 1.28·17-s + 1.02·18-s − 0.0277·19-s − 0.0309·20-s + 0.00669·21-s − 0.425·22-s − 1.52·23-s − 0.0173·24-s − 0.476·25-s + 1.44·26-s + 0.0354·27-s − 0.0161·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.11i)) \, \Gamma_{\R}(s+(1 - 4.11i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 7,\ (1 + 4.11900929292i, 1 - 4.11900929292i:\ ),\ -1)\)

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