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Properties

Label 1-4000-4000.1259-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.759 + 0.650i$
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  + (−0.612 + 0.790i)3-s + (−0.587 − 0.809i)7-s + (−0.248 − 0.968i)9-s + (−0.0941 + 0.995i)11-s + (0.509 − 0.860i)13-s + (−0.187 − 0.982i)17-s + (−0.790 + 0.612i)19-s + (0.999 + 0.0314i)21-s + (−0.844 − 0.535i)23-s + (0.917 + 0.397i)27-s + (−0.940 − 0.338i)29-s + (0.187 + 0.982i)31-s + (−0.728 − 0.684i)33-s + (−0.397 − 0.917i)37-s + (0.368 + 0.929i)39-s + ⋯
L(s)  = 1  + (−0.612 + 0.790i)3-s + (−0.587 − 0.809i)7-s + (−0.248 − 0.968i)9-s + (−0.0941 + 0.995i)11-s + (0.509 − 0.860i)13-s + (−0.187 − 0.982i)17-s + (−0.790 + 0.612i)19-s + (0.999 + 0.0314i)21-s + (−0.844 − 0.535i)23-s + (0.917 + 0.397i)27-s + (−0.940 − 0.338i)29-s + (0.187 + 0.982i)31-s + (−0.728 − 0.684i)33-s + (−0.397 − 0.917i)37-s + (0.368 + 0.929i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.759 + 0.650i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.759 + 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02902089636 - 0.07847472274i\)
\(L(\frac12)\) \(\approx\) \(0.02902089636 - 0.07847472274i\)
\(L(1)\) \(\approx\) \(0.6705199041 + 0.004469922731i\)
\(L(1)\) \(\approx\) \(0.6705199041 + 0.004469922731i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.612 + 0.790i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-0.0941 + 0.995i)T \)
13 \( 1 + (0.509 - 0.860i)T \)
17 \( 1 + (-0.187 - 0.982i)T \)
19 \( 1 + (-0.790 + 0.612i)T \)
23 \( 1 + (-0.844 - 0.535i)T \)
29 \( 1 + (-0.940 - 0.338i)T \)
31 \( 1 + (0.187 + 0.982i)T \)
37 \( 1 + (-0.397 - 0.917i)T \)
41 \( 1 + (0.844 - 0.535i)T \)
43 \( 1 + (0.891 - 0.453i)T \)
47 \( 1 + (-0.876 - 0.481i)T \)
53 \( 1 + (-0.999 - 0.0314i)T \)
59 \( 1 + (0.750 - 0.661i)T \)
61 \( 1 + (-0.218 - 0.975i)T \)
67 \( 1 + (0.338 + 0.940i)T \)
71 \( 1 + (0.481 - 0.876i)T \)
73 \( 1 + (-0.998 + 0.0627i)T \)
79 \( 1 + (-0.992 + 0.125i)T \)
83 \( 1 + (0.790 - 0.612i)T \)
89 \( 1 + (-0.998 + 0.0627i)T \)
97 \( 1 + (0.425 + 0.904i)T \)
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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

−18.71888372042507458136065703155, −18.18995285981433142898259116811, −17.36411340246462810685438649870, −16.719352067662878973170497026901, −16.12868621096729168426060632423, −15.489879025276716459643893347755, −14.57593492042375686638658633681, −13.78928168616315997482089366839, −13.02106637503323020289225361342, −12.79767106854572311227360948033, −11.708022719415918832737848457327, −11.37156012380812063187980537511, −10.68208131254483746591267659132, −9.69685246071025406874852836336, −8.85626070512199228566841079480, −8.32122754440015753081037353826, −7.52365573877746662413084977852, −6.47634971061866887363903492589, −6.14872797336945367701163562764, −5.62976534248480275421961550210, −4.55116850446633531747025498618, −3.68642425123859974521866412477, −2.692953183637965279928032266177, −1.960705781973626460695837914557, −1.13802476106281781723765560666, 0.02373952734128456047006309813, 0.52663643489102898955794557514, 1.766478980007675338702244754433, 2.86618565576483638726141737467, 3.76362199643500603416297308374, 4.22643130305337450209621212385, 5.07246147316280676544014381364, 5.82720428241618449629145866021, 6.57088348873708715254833111704, 7.25459064776759512843866561530, 8.09292285157634886174232754134, 9.08834943869786387625058337360, 9.73515496097939927315539414964, 10.364340647498327293513555943871, 10.773736814619059263883622735365, 11.614629120855601805997744476521, 12.612861004184022677963064036302, 12.770499537958756379262038259326, 13.95445498229095598410578454076, 14.51680428055491526549211014029, 15.36470151132677536940324600116, 16.02397377110820533286101993942, 16.32825964554487206509190315559, 17.38092429628909133560702822639, 17.60685605356532211650140696692

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