Properties

Degree $2$
Conductor $11$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 0.888·2-s − 1.51·3-s − 0.211·4-s + 0.374·5-s − 1.34·6-s − 0.686·7-s − 1.07·8-s + 1.28·9-s + 0.332·10-s + 0.301·11-s + 0.319·12-s − 0.897·13-s − 0.610·14-s − 0.566·15-s − 0.744·16-s + 1.66·17-s + 1.14·18-s − 1.91·19-s − 0.0792·20-s + 1.03·21-s + 0.267·22-s − 0.339·23-s + 1.62·24-s − 0.859·25-s − 0.796·26-s − 0.433·27-s + 0.145·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+4.55i) \, \Gamma_{\R}(s-4.55i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 11,\ (4.55526078532i, -4.55526078532i:\ ),\ -1)\)

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