Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.997 - 0.0674i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.0941 − 0.995i)3-s + (0.309 + 0.951i)7-s + (−0.982 − 0.187i)9-s + (−0.612 − 0.790i)11-s + (0.827 − 0.562i)13-s + (0.248 + 0.968i)17-s + (0.0941 + 0.995i)19-s + (0.975 − 0.218i)21-s + (0.728 + 0.684i)23-s + (−0.278 + 0.960i)27-s + (−0.750 − 0.661i)29-s + (0.968 − 0.248i)31-s + (−0.844 + 0.535i)33-s + (−0.960 + 0.278i)37-s + (−0.481 − 0.876i)39-s + ⋯
L(s,χ)  = 1  + (0.0941 − 0.995i)3-s + (0.309 + 0.951i)7-s + (−0.982 − 0.187i)9-s + (−0.612 − 0.790i)11-s + (0.827 − 0.562i)13-s + (0.248 + 0.968i)17-s + (0.0941 + 0.995i)19-s + (0.975 − 0.218i)21-s + (0.728 + 0.684i)23-s + (−0.278 + 0.960i)27-s + (−0.750 − 0.661i)29-s + (0.968 − 0.248i)31-s + (−0.844 + 0.535i)33-s + (−0.960 + 0.278i)37-s + (−0.481 − 0.876i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.997 - 0.0674i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.997 - 0.0674i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.997 - 0.0674i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (853, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ 0.997 - 0.0674i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.072330547 - 0.07001361824i$
$L(\frac12,\chi)$  $\approx$  $2.072330547 - 0.07001361824i$
$L(\chi,1)$  $\approx$  1.068000604 - 0.2242426056i
$L(1,\chi)$  $\approx$  1.068000604 - 0.2242426056i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.14330549880068178655303939198, −17.59125727405069350296007676456, −16.88177979336060131731639971796, −16.11644404223223542921475078425, −15.83367137625655610900375156330, −14.79852811705907770060943314443, −14.444950319558158529592743382859, −13.5059854969800535256083542186, −13.139946191435602686071852451537, −11.90987094797506747493097633649, −11.275831608858840759701855837936, −10.68895235030381064160992471846, −10.09054485105096782522438943928, −9.3501059553873735989230772035, −8.72053343313080056581796625703, −7.881401215603812067993310526715, −7.09058908861871886620198405592, −6.44572095955559797925390469494, −5.17656500240536706709705289938, −4.823102346595692324782791858581, −4.14755403401613021568220674803, −3.2630573104374369874041632925, −2.58220793468209168640382098660, −1.4246260446260195003022525795, −0.430267646388237033588362330902, 0.631688804132024835312793095890, 1.551561069954102369835586558645, 2.18467606001813911783227087240, 3.18133078000828287663085765467, 3.667625118508494480855781278096, 5.15383163660212633984861587310, 5.78357665944314916871522254583, 6.08188496104782519901180333545, 7.16236193666940409186798221716, 8.002214334833149372456434918153, 8.42172759327149541320746502669, 8.92881823498861619551227635452, 10.11508715374734589131640742242, 10.825392611583946955866836044633, 11.62192542417950234323020167437, 12.10799465150592708849961676209, 12.924569731367883050341833886803, 13.42506402935852572523829613116, 14.06267678047658572860665468779, 15.00818039586844381784142016560, 15.40468152125730988222883836747, 16.28609829449652495237834179309, 17.20115503682366459055015969168, 17.63677418226609736562497401564, 18.539011599034640318392989993390

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