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 + (0.360 − 0.356i)3-s + (−0.125 − 0.216i)4-s + (−0.592 − 0.583i)5-s + (0.0642 + 0.245i)6-s + (0.103 − 1.02i)7-s +(0.125)·8-s + (−0.357 − 0.613i)9-s + (0.400 − 0.110i)10-s + (1.34 − 0.122i)11-s + (−0.122 − 0.0334i)12-s + (0.0616 + 0.416i)13-s + (0.418 + 0.301i)14-s + (−0.421 + 0.000934i)15-s + (−0.0312 + 0.0541i)16-s + (−0.855 + 0.598i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+8.95i) \, \Gamma_{\R}(s+2.93i) \, \Gamma_{\R}(s-11.8i) \, L(s,f)\cr =\mathstrut & (-0.5 - 0.866i)\, \Lambda(1-s,\overline{f}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $-0.5 - 0.866i$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(3,\ 4,\ (8.9546625172i, 2.9365915306i, -11.8912540478i:\ ),\ -0.5 - 0.866i)$

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

−24.59912256, −22.57913504, −22.01745593, −20.29133986, −19.31006000, −17.95238412, −15.92737505, −14.40713480, −11.70886651, 0.93372346, 4.07774542, 6.85564032, 8.72396898, 14.26571491, 16.34014697, 17.69700217, 19.64960226, 20.25256015, 22.34994240, 23.85482092, 24.35540600

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