Properties

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

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Dirichlet series

$L(s,f)$  = 1  + 1.34·2-s − 0.577·3-s + 0.802·4-s − 0.0624·5-s − 0.775·6-s + 0.753·7-s − 0.265·8-s + 0.333·9-s − 0.0838·10-s − 0.408·11-s − 0.463·12-s + 0.527·13-s + 1.01·14-s + 0.0360·15-s − 1.15·16-s − 0.967·17-s + 0.447·18-s + 1.56·19-s − 0.0501·20-s − 0.435·21-s − 0.548·22-s + 0.285·23-s + 0.153·24-s − 0.996·25-s + 0.708·26-s − 0.192·27-s + 0.604·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+5.09i) \, \Gamma_{\R}(s-5.09i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 3,\ (5.09874190873i, -5.09874190873i:\ ),\ 1)$

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