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  + 1.13·2-s + 0.393·3-s + 0.289·4-s + 1.48·5-s + 0.447·6-s + 1.70·7-s − 0.807·8-s − 0.844·9-s + 1.69·10-s − 0.730·11-s + 0.113·12-s + 0.277·13-s + 1.93·14-s + 0.586·15-s − 1.20·16-s − 0.778·17-s − 0.959·18-s − 0.225·19-s + 0.430·20-s + 0.671·21-s − 0.829·22-s + 0.869·23-s − 0.317·24-s + 1.21·25-s + 0.314·26-s − 0.726·27-s + 0.493·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.96i)) \, \Gamma_{\R}(s+(1 - 4.96i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 13,\ (1 + 4.96964302847i, 1 - 4.96964302847i:\ ),\ 1)\)

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