Properties

Degree 3
Conductor $ 2^{2} $
Sign $-0.5 + 0.866i$
Primitive yes
Self-dual no
Analytic rank 0

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Dirichlet series

$L(s,f)$  = 1  + (−0.250 − 0.433i)2-s + (0.150 + 0.786i)3-s + (−0.125 + 0.216i)4-s + (1.15 − 0.148i)5-s + (0.303 − 0.261i)6-s + (−0.186 + 0.826i)7-s +(0.125)·8-s + (−0.746 + 1.02i)9-s + (−0.354 − 0.464i)10-s + (0.166 − 0.581i)11-s + (−0.189 − 0.0658i)12-s + (−0.265 + 0.00552i)13-s + (0.404 − 0.126i)14-s + (0.291 + 0.889i)15-s + (−0.0312 − 0.0541i)16-s + (−0.711 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+9.63i) \, \Gamma_{\R}(s+1.37i) \, \Gamma_{\R}(s-11.0i) \, L(s,f)\cr =\mathstrut & (-0.5 + 0.866i)\, \Lambda(1-s,\overline{f}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $-0.5 + 0.866i$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(3,\ 4,\ (9.63244453i, 1.3740602838i, -11.006504814i:\ ),\ -0.5 + 0.866i)$

Euler product

\[\begin{aligned} L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \overline{a(p)} p^{-2s} - p^{-3s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.827213, −23.759265, −22.556509, −20.850137, −19.503737, −17.825519, −17.197402, −14.699394, −13.175860, −6.562632, 2.532102, 5.347232, 9.173289, 13.859784, 16.054131, 17.551677, 19.159041, 20.684415, 21.753908, 22.426191, 24.786086

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