Properties

Degree 4
Conductor $ 1 $
Sign $1$
Primitive yes
Self-dual no
Analytic rank 0

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Dirichlet series

$L(s,f)$  = 1  + (0.556 + 0.928i)2-s + (−0.579 + 0.0446i)3-s + (0.179 + 1.03i)4-s + (0.652 − 0.534i)5-s + (−0.363 − 0.512i)6-s + (−0.493 − 0.0183i)7-s + (0.104 + 0.491i)8-s + (−0.572 − 0.0517i)9-s + (0.859 + 0.308i)10-s + (0.130 + 0.335i)11-s + (−0.149 − 0.590i)12-s + (0.259 − 0.340i)13-s + (−0.257 − 0.468i)14-s + (−0.354 + 0.338i)15-s + (−0.0961 + 1.12i)16-s + (0.282 − 0.246i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+16.8i) \, \Gamma_{\R}(s+2.27i) \, \Gamma_{\R}(s-6.03i) \, \Gamma_{\R}(s-13.1i) \, L(s,f)\cr =\mathstrut & \,\Lambda(1-s,\overline{f}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(4,\ 1,\ (16.89972715592i, 2.27258771492i, -6.03583588968i, -13.13647898116i:\ ),\ 1)$

Euler product

\[\begin{aligned} L(s,f) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−23.269117762, −22.422146922, −20.966010253, −19.295091037, −13.912595350, −11.648210792, −10.280168565, −6.028325425, 16.189015970, 17.414853804, 20.234411486, 21.920334804, 22.890264604, 24.278949288

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