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  − 0.629·2-s + 0.414·3-s − 0.603·4-s + 0.654·5-s − 0.261·6-s + 1.45·7-s + 1.00·8-s − 0.827·9-s − 0.411·10-s + 1.63·11-s − 0.250·12-s − 0.277·13-s − 0.917·14-s + 0.271·15-s − 0.0315·16-s + 0.371·17-s + 0.521·18-s + 0.566·19-s − 0.394·20-s + 0.604·21-s − 1.02·22-s + 0.464·23-s + 0.418·24-s − 0.572·25-s + 0.174·26-s − 0.758·27-s − 0.879·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+4.85i) \, \Gamma_{\R}(s-4.85i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 13,\ (4.8541487803i, -4.8541487803i:\ ),\ 1)\)

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