Properties

Degree 2
Conductor 3
Sign $-1$
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.671·2-s − 0.577·3-s − 0.548·4-s + 0.325·5-s + 0.387·6-s − 1.45·7-s + 1.04·8-s + 0.333·9-s − 0.218·10-s − 0.685·11-s + 0.317·12-s − 0.333·13-s + 0.977·14-s − 0.188·15-s − 0.149·16-s − 0.332·17-s − 0.223·18-s + 0.0618·19-s − 0.178·20-s + 0.840·21-s + 0.460·22-s − 0.154·23-s − 0.600·24-s − 0.893·25-s + 0.223·26-s − 0.192·27-s + 0.799·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+(1 + 6.12i)) \, \Gamma_{\R}(s+(1 - 6.12i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 3,\ (1 + 6.12057553309i, 1 - 6.12057553309i:\ ),\ -1)$

Euler product

\[\begin{aligned} L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line