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Properties

Label 1-4000-4000.773-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.455 - 0.890i$
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  + (−0.960 − 0.278i)3-s + (−0.809 − 0.587i)7-s + (0.844 + 0.535i)9-s + (0.397 + 0.917i)11-s + (0.975 − 0.218i)13-s + (0.684 + 0.728i)17-s + (−0.960 + 0.278i)19-s + (0.612 + 0.790i)21-s + (−0.637 + 0.770i)23-s + (−0.661 − 0.750i)27-s + (−0.827 − 0.562i)29-s + (0.728 − 0.684i)31-s + (−0.125 − 0.992i)33-s + (0.750 + 0.661i)37-s + (−0.998 − 0.0627i)39-s + ⋯
L(s)  = 1  + (−0.960 − 0.278i)3-s + (−0.809 − 0.587i)7-s + (0.844 + 0.535i)9-s + (0.397 + 0.917i)11-s + (0.975 − 0.218i)13-s + (0.684 + 0.728i)17-s + (−0.960 + 0.278i)19-s + (0.612 + 0.790i)21-s + (−0.637 + 0.770i)23-s + (−0.661 − 0.750i)27-s + (−0.827 − 0.562i)29-s + (0.728 − 0.684i)31-s + (−0.125 − 0.992i)33-s + (0.750 + 0.661i)37-s + (−0.998 − 0.0627i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.455 - 0.890i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.455 - 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7417077789 - 0.4537182042i\)
\(L(\frac12)\) \(\approx\) \(0.7417077789 - 0.4537182042i\)
\(L(1)\) \(\approx\) \(0.7086894565 - 0.03904722712i\)
\(L(1)\) \(\approx\) \(0.7086894565 - 0.03904722712i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.960 - 0.278i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.397 + 0.917i)T \)
13 \( 1 + (0.975 - 0.218i)T \)
17 \( 1 + (0.684 + 0.728i)T \)
19 \( 1 + (-0.960 + 0.278i)T \)
23 \( 1 + (-0.637 + 0.770i)T \)
29 \( 1 + (-0.827 - 0.562i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (0.750 + 0.661i)T \)
41 \( 1 + (-0.770 + 0.637i)T \)
43 \( 1 + (0.891 + 0.453i)T \)
47 \( 1 + (-0.904 - 0.425i)T \)
53 \( 1 + (-0.790 + 0.612i)T \)
59 \( 1 + (-0.509 - 0.860i)T \)
61 \( 1 + (-0.0941 - 0.995i)T \)
67 \( 1 + (-0.827 + 0.562i)T \)
71 \( 1 + (-0.904 - 0.425i)T \)
73 \( 1 + (0.968 - 0.248i)T \)
79 \( 1 + (0.876 - 0.481i)T \)
83 \( 1 + (0.278 + 0.960i)T \)
89 \( 1 + (-0.248 - 0.968i)T \)
97 \( 1 + (-0.982 + 0.187i)T \)
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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

−18.44687377640415114156341061861, −17.81547615372625195815719308569, −16.867638940106908734742494988073, −16.29009531340560928209178387490, −16.04778864159151099350563801689, −15.20221371586382676153998029021, −14.38484151367915204141331492208, −13.542394781249099098379505850652, −12.844277238023538517137531894742, −12.15667264390152179127621304446, −11.611373707779158713408172348034, −10.80790996976178205142421832616, −10.34499917165622790042472681600, −9.31219578896712956731245137311, −8.949044783358976255557634080529, −8.01456518044050685221209501932, −6.87847264486341106698950013520, −6.32951211647476440904825029247, −5.84038296798017289565816881972, −5.09524515611048056373770428322, −4.0922305059582606798208392184, −3.50364354373337995446222220145, −2.59485360773689172566012672301, −1.40232188018814275258960984256, −0.53967994253221511522816079821, 0.26020291809950178313401718185, 1.30822895055021000905953478396, 1.86671593582755329910594909547, 3.208434567402495956604278633744, 4.05161084255200520780944872975, 4.52238735476178621112451635013, 5.73161014890747028500986449875, 6.22155591437156575556548508624, 6.71788700787063986243942774100, 7.70940159061137459772311996808, 8.1445861094355079481285343934, 9.56096728139379615303110076740, 9.85649114453421693403162635134, 10.66905875567153577007530285972, 11.27400747116340472260984233401, 12.098432247576710749599403950129, 12.715253958613758289735850358482, 13.2368878446393763058342437252, 13.89158518256401488373863596579, 15.02705432959621788195753956008, 15.47597483931638885945299730010, 16.43142734330326463903006626967, 16.816962048540998490237605357642, 17.450619313383953912375642267765, 18.07118011183670593952298853449

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