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Properties

Label 1-4000-4000.1277-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.996 - 0.0800i$
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  + (−0.827 − 0.562i)3-s + (−0.809 − 0.587i)7-s + (0.368 + 0.929i)9-s + (0.509 − 0.860i)11-s + (0.397 − 0.917i)13-s + (0.481 − 0.876i)17-s + (−0.827 + 0.562i)19-s + (0.338 + 0.940i)21-s + (0.0627 − 0.998i)23-s + (0.218 − 0.975i)27-s + (0.612 + 0.790i)29-s + (0.876 + 0.481i)31-s + (−0.904 + 0.425i)33-s + (−0.975 + 0.218i)37-s + (−0.844 + 0.535i)39-s + ⋯
L(s)  = 1  + (−0.827 − 0.562i)3-s + (−0.809 − 0.587i)7-s + (0.368 + 0.929i)9-s + (0.509 − 0.860i)11-s + (0.397 − 0.917i)13-s + (0.481 − 0.876i)17-s + (−0.827 + 0.562i)19-s + (0.338 + 0.940i)21-s + (0.0627 − 0.998i)23-s + (0.218 − 0.975i)27-s + (0.612 + 0.790i)29-s + (0.876 + 0.481i)31-s + (−0.904 + 0.425i)33-s + (−0.975 + 0.218i)37-s + (−0.844 + 0.535i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.996 - 0.0800i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.996 - 0.0800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05644302039 - 1.408373473i\)
\(L(\frac12)\) \(\approx\) \(0.05644302039 - 1.408373473i\)
\(L(1)\) \(\approx\) \(0.7093486229 - 0.3861542922i\)
\(L(1)\) \(\approx\) \(0.7093486229 - 0.3861542922i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.827 - 0.562i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.509 - 0.860i)T \)
13 \( 1 + (0.397 - 0.917i)T \)
17 \( 1 + (0.481 - 0.876i)T \)
19 \( 1 + (-0.827 + 0.562i)T \)
23 \( 1 + (0.0627 - 0.998i)T \)
29 \( 1 + (0.612 + 0.790i)T \)
31 \( 1 + (0.876 + 0.481i)T \)
37 \( 1 + (-0.975 + 0.218i)T \)
41 \( 1 + (-0.998 + 0.0627i)T \)
43 \( 1 + (0.453 - 0.891i)T \)
47 \( 1 + (0.684 - 0.728i)T \)
53 \( 1 + (0.940 - 0.338i)T \)
59 \( 1 + (-0.995 - 0.0941i)T \)
61 \( 1 + (0.750 - 0.661i)T \)
67 \( 1 + (0.612 - 0.790i)T \)
71 \( 1 + (0.684 - 0.728i)T \)
73 \( 1 + (-0.637 + 0.770i)T \)
79 \( 1 + (-0.187 - 0.982i)T \)
83 \( 1 + (-0.562 - 0.827i)T \)
89 \( 1 + (-0.770 - 0.637i)T \)
97 \( 1 + (0.125 - 0.992i)T \)
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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

−18.724924353636809207766430591602, −17.736007800040710921387635985865, −17.21500806209758802850106475986, −16.73084187445971162653821328782, −15.80775862377666058389144760385, −15.42275301856567268744402299953, −14.82989496411473251931184592675, −13.85222412579284081526803466373, −13.01126252294711727386143506692, −12.29667345463573332002638747218, −11.85053113529974316218228871713, −11.139235297976567352980162951442, −10.248095687784372309465903651451, −9.72400795151360638449316280228, −9.11711228292991701608138611517, −8.38712373988443457955706016754, −7.150175137445868764856888697884, −6.54032236039141177363112881214, −6.02121200927955840466394727244, −5.24877728179712010985175067398, −4.22444627282793825268529054877, −3.94019864437343563575625498648, −2.82656944966042273603209601649, −1.831111859139392459464628994601, −0.90320485177265946679430367817, 0.40033012073226376698500678925, 0.68876123402543280949151719296, 1.71740718156230962370371941029, 2.90876711999711250000775511971, 3.52362722722948233361709794882, 4.51365393830859720937283953987, 5.37200791807375156637066196800, 6.06058935648193375927432493972, 6.705569681905962866642709740140, 7.201104249705228226359545875534, 8.266509171212435165496597544091, 8.72038715090614239713661423613, 9.99749242466937187556252155660, 10.443277198233987671277239425826, 10.98485807200087355975139372935, 12.01838246008497869219943459015, 12.36995628272654332277969115620, 13.16941694975390484701332930487, 13.794829293759280770382849487562, 14.31642786625859180272498404915, 15.56690823118019546249659607496, 16.040457672404308448240465110547, 16.89533875569494077281129092174, 17.01146944227657002639301821820, 18.041294366013680728975047602132

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