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Properties

Label 1-6001-6001.180-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.709 - 0.705i$
Analytic cond. $27.8685$
Root an. cond. $27.8685$
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  + (−0.540 + 0.841i)2-s + (0.999 + 0.0178i)3-s + (−0.415 − 0.909i)4-s + (0.719 − 0.694i)5-s + (−0.555 + 0.831i)6-s + (0.195 − 0.980i)7-s + (0.989 + 0.142i)8-s + (0.999 + 0.0356i)9-s + (0.195 + 0.980i)10-s + (0.106 + 0.994i)11-s + (−0.399 − 0.916i)12-s + (−0.992 − 0.124i)13-s + (0.719 + 0.694i)14-s + (0.731 − 0.681i)15-s + (−0.654 + 0.755i)16-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)2-s + (0.999 + 0.0178i)3-s + (−0.415 − 0.909i)4-s + (0.719 − 0.694i)5-s + (−0.555 + 0.831i)6-s + (0.195 − 0.980i)7-s + (0.989 + 0.142i)8-s + (0.999 + 0.0356i)9-s + (0.195 + 0.980i)10-s + (0.106 + 0.994i)11-s + (−0.399 − 0.916i)12-s + (−0.992 − 0.124i)13-s + (0.719 + 0.694i)14-s + (0.731 − 0.681i)15-s + (−0.654 + 0.755i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.709 - 0.705i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.709 - 0.705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3131172353 - 0.7589814993i\)
\(L(\frac12)\) \(\approx\) \(0.3131172353 - 0.7589814993i\)
\(L(1)\) \(\approx\) \(1.030033247 + 0.01460302121i\)
\(L(1)\) \(\approx\) \(1.030033247 + 0.01460302121i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.540 + 0.841i)T \)
3 \( 1 + (0.999 + 0.0178i)T \)
5 \( 1 + (0.719 - 0.694i)T \)
7 \( 1 + (0.195 - 0.980i)T \)
11 \( 1 + (0.106 + 0.994i)T \)
13 \( 1 + (-0.992 - 0.124i)T \)
19 \( 1 + (-0.315 - 0.948i)T \)
23 \( 1 + (-0.281 - 0.959i)T \)
29 \( 1 + (-0.681 - 0.731i)T \)
31 \( 1 + (-0.996 - 0.0891i)T \)
37 \( 1 + (-0.902 + 0.431i)T \)
41 \( 1 + (-0.281 + 0.959i)T \)
43 \( 1 + (-0.627 + 0.778i)T \)
47 \( 1 + (-0.999 + 0.0356i)T \)
53 \( 1 + (-0.0178 + 0.999i)T \)
59 \( 1 + (-0.555 + 0.831i)T \)
61 \( 1 + (0.315 - 0.948i)T \)
67 \( 1 + (-0.831 + 0.555i)T \)
71 \( 1 + (0.973 - 0.229i)T \)
73 \( 1 + (0.0356 + 0.999i)T \)
79 \( 1 + (-0.719 - 0.694i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (-0.668 - 0.743i)T \)
97 \( 1 + (-0.510 - 0.860i)T \)
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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

−18.27114567617714627371348816788, −17.56624774859122124608274713026, −16.76190623737531903039898108785, −16.08575368387765177086519695346, −15.147294768544278571010149714690, −14.55292892843682477263935961310, −14.0030586358528529773214505173, −13.40412921694076130947072186124, −12.60405035309325078450554895566, −12.09986781584287480126707800460, −11.2157687781574650361179864712, −10.605664287157144297347607653788, −9.82595136516885432326167838519, −9.39937287633443442249474291139, −8.75327074745849670494883636312, −8.18227116673652937319910134161, −7.38845286564366694859988509192, −6.761418170297846279136498120010, −5.62779981102701493214849403352, −5.06032088931027845638268644408, −3.661551231814261532562202127446, −3.4513400411053191549072899676, −2.58779836931639254993269907628, −1.893497677878277903276265112940, −1.602308109221823871663504735903, 0.18119345395382574707361601445, 1.37234009603655109661950359730, 1.87249055166270072986057525511, 2.71267751498857314765466750002, 4.0244371069728538975214492129, 4.7237875630225891551673171326, 4.92966183147105568178393559244, 6.20678470642775145753486132105, 6.902590263199690529864171522562, 7.45693580428200160811673027682, 8.044209639643733513967613614856, 8.81262553121644384766784970887, 9.375124635401137291821563380300, 10.09540939423350168078542766090, 10.207849503555875594953105116507, 11.384243627022694263440709640188, 12.641908192250865981952073495636, 13.03964546420261147443038284471, 13.68299320150602267772425007105, 14.38748827444454557574351136607, 14.77840040222048703557696395926, 15.43559611613162533890240074626, 16.25586775823867093582716815849, 16.96824267940293951615282377233, 17.26492936221666725903801944423

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