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Properties

Label 1-4000-4000.853-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.997 - 0.0674i$
Analytic cond. $429.859$
Root an. cond. $429.859$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.997 - 0.0674i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.997 - 0.0674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.072330547 - 0.07001361824i\)
\(L(\frac12)\) \(\approx\) \(2.072330547 - 0.07001361824i\)
\(L(1)\) \(\approx\) \(1.068000604 - 0.2242426056i\)
\(L(1)\) \(\approx\) \(1.068000604 - 0.2242426056i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.0941 - 0.995i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.612 - 0.790i)T \)
13 \( 1 + (0.827 - 0.562i)T \)
17 \( 1 + (0.248 + 0.968i)T \)
19 \( 1 + (0.0941 + 0.995i)T \)
23 \( 1 + (0.728 + 0.684i)T \)
29 \( 1 + (-0.750 - 0.661i)T \)
31 \( 1 + (0.968 - 0.248i)T \)
37 \( 1 + (-0.960 + 0.278i)T \)
41 \( 1 + (-0.684 - 0.728i)T \)
43 \( 1 + (0.156 - 0.987i)T \)
47 \( 1 + (-0.368 - 0.929i)T \)
53 \( 1 + (0.218 + 0.975i)T \)
59 \( 1 + (0.338 - 0.940i)T \)
61 \( 1 + (0.999 - 0.0314i)T \)
67 \( 1 + (-0.750 + 0.661i)T \)
71 \( 1 + (-0.368 - 0.929i)T \)
73 \( 1 + (-0.425 + 0.904i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (0.995 - 0.0941i)T \)
89 \( 1 + (0.904 + 0.425i)T \)
97 \( 1 + (0.998 - 0.0627i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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