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  + 1.04·2-s + 1.84·3-s + 0.100·4-s + 0.0514·5-s + 1.93·6-s − 0.0890·7-s − 0.943·8-s + 2.41·9-s + 0.0539·10-s − 1.40·11-s + 0.185·12-s − 0.277·13-s − 0.0934·14-s + 0.0951·15-s − 1.09·16-s + 1.01·17-s + 2.53·18-s − 0.300·19-s + 0.00515·20-s − 0.164·21-s − 1.47·22-s + 0.551·23-s − 1.74·24-s − 0.997·25-s − 0.290·26-s + 2.61·27-s − 0.00892·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+4.46i) \, \Gamma_{\R}(s-4.46i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 13,\ (4.46679817957i, -4.46679817957i:\ ),\ 1)\)

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