Properties

Degree $2$
Conductor $28$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.991·3-s + 1.45·5-s + 0.377·7-s − 0.0172·9-s − 0.663·11-s + 1.59·13-s − 1.44·15-s − 0.979·17-s + 0.0817·19-s − 0.374·21-s + 0.300·23-s + 1.11·25-s + 1.00·27-s − 0.734·29-s + 0.501·31-s + 0.657·33-s + 0.549·35-s − 0.915·37-s − 1.57·39-s − 1.12·41-s + 0.613·43-s − 0.0250·45-s − 0.182·47-s + 0.142·49-s + 0.971·51-s + 0.402·53-s − 0.963·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s+1.69i) \, \Gamma_{\R}(s-1.69i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 28,\ (1.69299215834i, -1.69299215834i:\ ),\ 1)\)

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