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  + 0.623·2-s − 1.58·3-s − 0.610·4-s + 0.694·5-s − 0.986·6-s − 0.377·7-s − 1.00·8-s + 1.49·9-s + 0.433·10-s − 1.40·11-s + 0.965·12-s + 0.0487·13-s − 0.235·14-s − 1.09·15-s − 0.0159·16-s + 0.192·17-s + 0.935·18-s + 0.550·19-s − 0.424·20-s + 0.597·21-s − 0.879·22-s − 1.16·23-s + 1.58·24-s − 0.517·25-s + 0.0304·26-s − 0.790·27-s + 0.230·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s+(1 + 5.10i)) \, \Gamma_{\R}(s+(1 - 5.10i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 7,\ (1 + 5.10146081193i, 1 - 5.10146081193i:\ ),\ -1)\)

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