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  − 0.0483·2-s + 0.992·3-s − 0.997·4-s − 0.646·5-s − 0.0480·6-s − 1.21·7-s + 0.0966·8-s − 0.0145·9-s + 0.0312·10-s − 0.301·11-s − 0.990·12-s − 0.206·13-s + 0.0588·14-s − 0.641·15-s + 0.992·16-s + 1.39·17-s + 0.000704·18-s − 1.28·19-s + 0.645·20-s − 1.20·21-s + 0.0145·22-s + 0.537·23-s + 0.0959·24-s − 0.581·25-s + 0.00997·26-s − 1.00·27-s + 1.21·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+(1 + 5.38i)) \, \Gamma_{\R}(s+(1 - 5.38i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 11,\ (1 + 5.38428160735i, 1 - 5.38428160735i:\ ),\ -1)\)

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