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  + 1.59·2-s + 0.855·3-s + 1.53·4-s + 0.280·5-s + 1.36·6-s + 1.13·7-s + 0.854·8-s − 0.268·9-s + 0.446·10-s − 0.301·11-s + 1.31·12-s + 0.598·13-s + 1.81·14-s + 0.239·15-s − 0.175·16-s − 0.285·17-s − 0.427·18-s + 1.74·19-s + 0.430·20-s + 0.974·21-s − 0.480·22-s − 0.626·23-s + 0.731·24-s − 0.921·25-s + 0.952·26-s − 1.08·27-s + 1.75·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+8.14i) \, \Gamma_{\R}(s-8.14i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 11,\ (8.14202179525i, -8.14202179525i:\ ),\ 1)\)

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