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  − 1.49·2-s − 1.47·3-s + 1.23·4-s − 0.910·5-s + 2.20·6-s − 1.39·7-s − 0.354·8-s + 1.16·9-s + 1.36·10-s + 0.301·11-s − 1.82·12-s − 1.18·13-s + 2.08·14-s + 1.34·15-s − 0.706·16-s − 0.441·17-s − 1.74·18-s + 0.570·19-s − 1.12·20-s + 2.05·21-s − 0.450·22-s + 0.594·23-s + 0.522·24-s − 0.170·25-s + 1.76·26-s − 0.246·27-s − 1.72·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.96i)) \, \Gamma_{\R}(s+(1 - 4.96i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 11,\ (1 + 4.96681056895i, 1 - 4.96681056895i:\ ),\ 1)\)

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