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  + 1.52·2-s + 0.417·3-s + 1.32·4-s + 1.27·5-s + 0.637·6-s − 0.976·7-s + 0.494·8-s − 0.825·9-s + 1.95·10-s + 0.301·11-s + 0.553·12-s − 0.370·13-s − 1.48·14-s + 0.534·15-s − 0.570·16-s − 0.0675·17-s − 1.25·18-s + 1.65·19-s + 1.69·20-s − 0.407·21-s + 0.459·22-s − 1.37·23-s + 0.206·24-s + 0.635·25-s − 0.565·26-s − 0.762·27-s − 1.29·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.73i)) \, \Gamma_{\R}(s+(1 - 4.73i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 11,\ (1 + 4.73801385946i, 1 - 4.73801385946i:\ ),\ 1)\)

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