Properties

Degree $2$
Conductor $13$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 0.639·2-s − 1.02·3-s − 0.591·4-s + 0.274·5-s − 0.652·6-s − 0.609·7-s − 1.01·8-s + 0.0425·9-s + 0.175·10-s + 0.501·11-s + 0.603·12-s + 0.277·13-s − 0.389·14-s − 0.279·15-s − 0.0592·16-s − 1.81·17-s + 0.0272·18-s + 0.964·19-s − 0.162·20-s + 0.622·21-s + 0.320·22-s − 0.669·23-s + 1.03·24-s − 0.924·25-s + 0.177·26-s + 0.977·27-s + 0.360·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+4.23i) \, \Gamma_{\R}(s-4.23i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 13,\ (4.23019956539i, -4.23019956539i:\ ),\ -1)\)

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