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  + 1.82·2-s + 0.840·3-s + 2.32·4-s − 1.46·5-s + 1.53·6-s + 0.112·7-s + 2.42·8-s − 0.293·9-s − 2.66·10-s − 0.301·11-s + 1.95·12-s − 0.323·13-s + 0.205·14-s − 1.22·15-s + 2.09·16-s + 1.15·17-s − 0.535·18-s + 0.199·19-s − 3.40·20-s + 0.0945·21-s − 0.550·22-s + 0.593·23-s + 2.03·24-s + 1.13·25-s − 0.589·26-s − 1.08·27-s + 0.261·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+4.71i) \, \Gamma_{\R}(s-4.71i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 11,\ (4.71167698681i, -4.71167698681i:\ ),\ 1)\)

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