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Properties

Label 1-4000-4000.2237-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.452 + 0.891i$
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 + (−0.809 + 0.587i)7-s + (−0.248 − 0.968i)9-s + (−0.995 − 0.0941i)11-s + (−0.509 + 0.860i)13-s + (−0.982 + 0.187i)17-s + (−0.612 − 0.790i)19-s + (0.0314 − 0.999i)21-s + (0.535 − 0.844i)23-s + (0.917 + 0.397i)27-s + (−0.338 + 0.940i)29-s + (−0.187 − 0.982i)31-s + (0.684 − 0.728i)33-s + (0.397 + 0.917i)37-s + (−0.368 − 0.929i)39-s + ⋯
L(s)  = 1  + (−0.612 + 0.790i)3-s + (−0.809 + 0.587i)7-s + (−0.248 − 0.968i)9-s + (−0.995 − 0.0941i)11-s + (−0.509 + 0.860i)13-s + (−0.982 + 0.187i)17-s + (−0.612 − 0.790i)19-s + (0.0314 − 0.999i)21-s + (0.535 − 0.844i)23-s + (0.917 + 0.397i)27-s + (−0.338 + 0.940i)29-s + (−0.187 − 0.982i)31-s + (0.684 − 0.728i)33-s + (0.397 + 0.917i)37-s + (−0.368 − 0.929i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.452 + 0.891i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (2237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.452 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09104746135 + 0.1483143572i\)
\(L(\frac12)\) \(\approx\) \(0.09104746135 + 0.1483143572i\)
\(L(1)\) \(\approx\) \(0.5376693751 + 0.1302573853i\)
\(L(1)\) \(\approx\) \(0.5376693751 + 0.1302573853i\)

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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.995 - 0.0941i)T \)
13 \( 1 + (-0.509 + 0.860i)T \)
17 \( 1 + (-0.982 + 0.187i)T \)
19 \( 1 + (-0.612 - 0.790i)T \)
23 \( 1 + (0.535 - 0.844i)T \)
29 \( 1 + (-0.338 + 0.940i)T \)
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 \( 1 + (0.481 - 0.876i)T \)
53 \( 1 + (-0.999 - 0.0314i)T \)
59 \( 1 + (-0.661 - 0.750i)T \)
61 \( 1 + (0.975 - 0.218i)T \)
67 \( 1 + (-0.338 - 0.940i)T \)
71 \( 1 + (0.481 - 0.876i)T \)
73 \( 1 + (0.0627 + 0.998i)T \)
79 \( 1 + (-0.992 + 0.125i)T \)
83 \( 1 + (0.790 - 0.612i)T \)
89 \( 1 + (-0.998 + 0.0627i)T \)
97 \( 1 + (0.904 - 0.425i)T \)
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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

−18.0480136569855226179763268954, −17.34934613449065298362954731684, −16.99366991412316045245344933626, −15.998732016129299988345809591258, −15.62477822972295711511543167025, −14.65796522982710270281763626926, −13.69540830424863762818290663624, −13.16415123520484718434142488363, −12.76096148931484640452164035907, −12.06475470485777227054446623716, −11.13660504546756241210865186994, −10.51818955976196276272889545094, −10.04609099231169080563330605845, −9.03161222724106034855882943354, −8.07377317681540978129007318802, −7.48108742482729612184062309410, −6.92006535733918831618546250125, −6.10445136207387987026401090081, −5.45140474200071563498633609838, −4.72630026983231627610134347108, −3.703953030432577078670128087543, −2.774127960312912485750611011675, −2.087929859387364523882518265265, −1.012480410648022917729265769410, −0.08291724501247618798716716128, 0.318138093572954847106190789021, 1.91664848074899465188408707426, 2.74435078319067341399261315772, 3.43631291424996007405222877947, 4.58682396101009360053763947950, 4.84423724191599432526915658050, 5.81904521344169117116196568463, 6.56409446673797126300115164803, 6.98582503428166379381651705537, 8.34616517559337799267710697114, 8.90823617198474382523605453476, 9.62630167798609364845216584945, 10.17607897232075647605232320988, 11.08625495134669546218846357734, 11.41412599181742837128548978085, 12.45889222829467412322742682452, 12.8971392314327273379209080066, 13.65539673199786604068990978549, 14.80261074753373928803260461501, 15.24141968540534837511690351202, 15.759709598222434561621398881283, 16.69491154015167226492616001765, 16.80874942679109317569480757764, 17.871865939330791320784485330205, 18.52400361211660420872885008188

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