Properties

Label 1-4000-4000.317-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  + (0.975 + 0.218i)3-s + (0.309 − 0.951i)7-s + (0.904 + 0.425i)9-s + (0.0314 − 0.999i)11-s + (−0.338 − 0.940i)13-s + (0.998 − 0.0627i)17-s + (0.975 − 0.218i)19-s + (0.509 − 0.860i)21-s + (−0.187 + 0.982i)23-s + (0.790 + 0.612i)27-s + (0.397 + 0.917i)29-s + (0.0627 + 0.998i)31-s + (0.248 − 0.968i)33-s + (0.612 + 0.790i)37-s + (−0.125 − 0.992i)39-s + ⋯
L(s)  = 1  + (0.975 + 0.218i)3-s + (0.309 − 0.951i)7-s + (0.904 + 0.425i)9-s + (0.0314 − 0.999i)11-s + (−0.338 − 0.940i)13-s + (0.998 − 0.0627i)17-s + (0.975 − 0.218i)19-s + (0.509 − 0.860i)21-s + (−0.187 + 0.982i)23-s + (0.790 + 0.612i)27-s + (0.397 + 0.917i)29-s + (0.0627 + 0.998i)31-s + (0.248 − 0.968i)33-s + (0.612 + 0.790i)37-s + (−0.125 − 0.992i)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: $\chi_{4000} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.997 + 0.0674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.701083664 + 0.1588259543i\)
\(L(\frac12)\) \(\approx\) \(4.701083664 + 0.1588259543i\)
\(L(1)\) \(\approx\) \(1.786259648 - 0.06838180918i\)
\(L(1)\) \(\approx\) \(1.786259648 - 0.06838180918i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.975 + 0.218i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.0314 - 0.999i)T \)
13 \( 1 + (-0.338 - 0.940i)T \)
17 \( 1 + (0.998 - 0.0627i)T \)
19 \( 1 + (0.975 - 0.218i)T \)
23 \( 1 + (-0.187 + 0.982i)T \)
29 \( 1 + (0.397 + 0.917i)T \)
31 \( 1 + (0.0627 + 0.998i)T \)
37 \( 1 + (0.612 + 0.790i)T \)
41 \( 1 + (0.982 - 0.187i)T \)
43 \( 1 + (0.156 + 0.987i)T \)
47 \( 1 + (0.770 + 0.637i)T \)
53 \( 1 + (-0.860 - 0.509i)T \)
59 \( 1 + (0.278 - 0.960i)T \)
61 \( 1 + (-0.827 + 0.562i)T \)
67 \( 1 + (0.397 - 0.917i)T \)
71 \( 1 + (0.770 + 0.637i)T \)
73 \( 1 + (0.876 + 0.481i)T \)
79 \( 1 + (0.535 + 0.844i)T \)
83 \( 1 + (0.218 + 0.975i)T \)
89 \( 1 + (-0.481 + 0.876i)T \)
97 \( 1 + (0.368 - 0.929i)T \)
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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

−18.51167947306016634908371258420, −17.83208727846045315375776682238, −16.952142271199498472895627447047, −16.12563431422826440449045027285, −15.43270863075389813426294233560, −14.8000496977250016354749681748, −14.31763276537495505717174538762, −13.71100616705804721385663091223, −12.71338027218420374926102369883, −12.167032774897958833569219457804, −11.767099139122950833787012725703, −10.582204764066356454206784331217, −9.5699794044686582782253336534, −9.48195461956274475125556488830, −8.55848550962190356090486150866, −7.75816878120642161215005313297, −7.36087106979458009866084030727, −6.37890677013843238758900094851, −5.6096423930905986122907774619, −4.58150479014492852529383797194, −4.07413630044354087234518322049, −2.9880506512908424304578781446, −2.27205533233682550918531092855, −1.79146303422262027534967835400, −0.67912366876628313504544823642, 0.95785057416592347098045627327, 1.214763128062317247018552715986, 2.62110404815737804986673574999, 3.31573705541490975631628877844, 3.6980564659852988567648488728, 4.85126058791013846226610621831, 5.35077222943813193680189462515, 6.462783263864512556444983796756, 7.43637756374292074236498441441, 7.83521481887832393795308236226, 8.40391774071512216248387704096, 9.4556368406083019689533382619, 9.859855864241184483377154068404, 10.723266640423238862684501528890, 11.20684783967939669133973345115, 12.32463780544469136552611302354, 12.96485395731256303707715327631, 13.90513758391950424258621071512, 14.021681149825330870365555155645, 14.76164816820228911254562358530, 15.687285158259024751826694918009, 16.14695650732158235989042926888, 16.87485172083850434360326390644, 17.718162900412311305013956023689, 18.3357899381143918017833175931

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