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  − 0.535·2-s + 0.385·3-s − 0.713·4-s − 0.959·5-s − 0.206·6-s − 0.213·7-s + 0.917·8-s − 0.851·9-s + 0.513·10-s + 0.301·11-s − 0.275·12-s − 0.455·13-s + 0.114·14-s − 0.370·15-s + 0.222·16-s − 1.21·17-s + 0.455·18-s + 0.988·19-s + 0.684·20-s − 0.0824·21-s − 0.161·22-s − 0.271·23-s + 0.353·24-s − 0.0792·25-s + 0.243·26-s − 0.714·27-s + 0.152·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+4.01i) \, \Gamma_{\R}(s-4.01i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 11,\ (4.01806918822i, -4.01806918822i:\ ),\ -1)\)

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