Properties

Degree $2$
Conductor $14$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 0.707·2-s + 0.562·3-s + 0.5·4-s + 1.61·5-s + 0.397·6-s − 0.377·7-s + 0.353·8-s − 0.683·9-s + 1.14·10-s − 0.570·11-s + 0.281·12-s + 1.01·13-s − 0.267·14-s + 0.907·15-s + 0.250·16-s − 1.54·17-s − 0.483·18-s − 0.779·19-s + 0.806·20-s − 0.212·21-s − 0.403·22-s + 0.274·23-s + 0.198·24-s + 1.60·25-s + 0.715·26-s − 0.947·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 14 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.29i)) \, \Gamma_{\R}(s+(1 - 4.29i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 14,\ (1 + 4.29731218878i, 1 - 4.29731218878i:\ ),\ 1)\)

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