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  − 1.68·2-s − 0.549·3-s + 1.84·4-s − 1.41·5-s + 0.926·6-s + 1.07·7-s − 1.42·8-s − 0.698·9-s + 2.37·10-s + 0.860·11-s − 1.01·12-s + 0.277·13-s − 1.80·14-s + 0.774·15-s + 0.559·16-s − 0.293·17-s + 1.17·18-s + 1.11·19-s − 2.60·20-s − 0.587·21-s − 1.45·22-s + 1.30·23-s + 0.783·24-s + 0.988·25-s − 0.467·26-s + 0.932·27-s + 1.97·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.38i)) \, \Gamma_{\R}(s+(1 - 4.38i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 13,\ (1 + 4.38518054954i, 1 - 4.38518054954i:\ ),\ 1)\)

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