Properties

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

Related objects

Nearby objects

next$L(s,f_{\text next})$

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Dirichlet series

$L(s,f)$  = 1  + 0.272·2-s + 0.577·3-s − 0.925·4-s + 0.696·5-s + 0.157·6-s − 0.241·7-s − 0.525·8-s + 0.333·9-s + 0.190·10-s + 0.613·11-s − 0.534·12-s − 0.649·13-s − 0.0660·14-s + 0.401·15-s + 0.782·16-s − 0.285·17-s + 0.0909·18-s − 0.381·19-s − 0.644·20-s − 0.139·21-s + 0.167·22-s + 1.52·23-s − 0.303·24-s − 0.515·25-s − 0.177·26-s + 0.192·27-s + 0.223·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.38i)) \, \Gamma_{\R}(s+(1 - 4.38i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 3,\ (1 + 4.38805356322i, 1 - 4.38805356322i:\ ),\ 1)$

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