Properties

Degree $2$
Conductor $7$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

Learn more

Dirichlet series

$L(s,f)$  = 1  + 1.72·2-s + 1.68·3-s + 1.97·4-s − 0.388·5-s + 2.90·6-s + 0.377·7-s + 1.67·8-s + 1.84·9-s − 0.669·10-s − 1.34·11-s + 3.32·12-s − 0.897·13-s + 0.651·14-s − 0.654·15-s + 0.913·16-s + 0.555·17-s + 3.17·18-s − 0.416·19-s − 0.765·20-s + 0.637·21-s − 2.31·22-s + 1.62·23-s + 2.82·24-s − 0.849·25-s − 1.54·26-s + 1.41·27-s + 0.744·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s+(1 + 10.6i)) \, \Gamma_{\R}(s+(1 - 10.6i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 7,\ (1 + 10.6717085945i, 1 - 10.6717085945i:\ ),\ 1)\)

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