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Properties

Label 1-6001-6001.1006-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.983 - 0.180i$
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  + (0.841 + 0.540i)2-s + (0.0178 − 0.999i)3-s + (0.415 + 0.909i)4-s + (−0.694 − 0.719i)5-s + (0.555 − 0.831i)6-s + (0.980 + 0.195i)7-s + (−0.142 + 0.989i)8-s + (−0.999 − 0.0356i)9-s + (−0.195 − 0.980i)10-s + (0.627 − 0.778i)11-s + (0.916 − 0.399i)12-s + (0.789 − 0.613i)13-s + (0.719 + 0.694i)14-s + (−0.731 + 0.681i)15-s + (−0.654 + 0.755i)16-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (0.0178 − 0.999i)3-s + (0.415 + 0.909i)4-s + (−0.694 − 0.719i)5-s + (0.555 − 0.831i)6-s + (0.980 + 0.195i)7-s + (−0.142 + 0.989i)8-s + (−0.999 − 0.0356i)9-s + (−0.195 − 0.980i)10-s + (0.627 − 0.778i)11-s + (0.916 − 0.399i)12-s + (0.789 − 0.613i)13-s + (0.719 + 0.694i)14-s + (−0.731 + 0.681i)15-s + (−0.654 + 0.755i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.983 - 0.180i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (1006, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.983 - 0.180i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.415199580 - 0.3100769421i\)
\(L(\frac12)\) \(\approx\) \(3.415199580 - 0.3100769421i\)
\(L(1)\) \(\approx\) \(1.827305833 - 0.07417923090i\)
\(L(1)\) \(\approx\) \(1.827305833 - 0.07417923090i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (0.0178 - 0.999i)T \)
5 \( 1 + (-0.694 - 0.719i)T \)
7 \( 1 + (0.980 + 0.195i)T \)
11 \( 1 + (0.627 - 0.778i)T \)
13 \( 1 + (0.789 - 0.613i)T \)
19 \( 1 + (0.315 + 0.948i)T \)
23 \( 1 + (0.877 + 0.479i)T \)
29 \( 1 + (-0.999 - 0.0356i)T \)
31 \( 1 + (0.0891 - 0.996i)T \)
37 \( 1 + (0.902 - 0.431i)T \)
41 \( 1 + (0.479 + 0.877i)T \)
43 \( 1 + (-0.627 + 0.778i)T \)
47 \( 1 + (-0.0356 - 0.999i)T \)
53 \( 1 + (0.694 + 0.719i)T \)
59 \( 1 + (-0.195 - 0.980i)T \)
61 \( 1 + (0.894 - 0.447i)T \)
67 \( 1 + (-0.195 + 0.980i)T \)
71 \( 1 + (0.973 - 0.229i)T \)
73 \( 1 + (0.731 + 0.681i)T \)
79 \( 1 + (-0.694 + 0.719i)T \)
83 \( 1 + (-0.989 + 0.142i)T \)
89 \( 1 + (0.998 + 0.0535i)T \)
97 \( 1 + (0.968 + 0.247i)T \)
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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

−17.81158214013132718391045891586, −16.940624224076106530164918078952, −16.199808482574003342199343178698, −15.45236166712168713982697497475, −15.05379892864587914296467236874, −14.49206838639978355242333997085, −14.05123197553037602944181321750, −13.26468167443868063672014480426, −12.20954366423847458154231909995, −11.60501274362929749649622230963, −11.162355863740544302373501574425, −10.743632134687071426051787254029, −10.02429127684758326221551282253, −9.1458205657987889217604657238, −8.62264247337362434226650947383, −7.45338112349612036673355792066, −6.892153743029376204139111678439, −6.11747333575271451601821002867, −5.12824125922409913291038553213, −4.61296436447421095328831763769, −4.03964834374209219046052858306, −3.48490555637041586387562024896, −2.66234105734761411809442922197, −1.87261947067813958984423605900, −0.82497213504335637540299008457, 0.87576544209896236295716874439, 1.48763653067260332360159783430, 2.46871604964943827435847220125, 3.48203254029895326355534802018, 3.89062573313125703451145348577, 4.9141044589507509470065396033, 5.63042639067548769988363920874, 5.97112828728375735723529301373, 6.95560863859042876484164514682, 7.74257011088523572273159897877, 8.09347153907494657268794109400, 8.58608354515924930729683596626, 9.35752727406174874891242175320, 11.0213790448218294834438070959, 11.40873369020120378140744504830, 11.750629841576697523154718705135, 12.64345547242282776982261680314, 13.10970815395434304668220598258, 13.64764931939700903839091074454, 14.50974985768231181444956304646, 14.86633409942710558223545343384, 15.65362782911931292153464094987, 16.48271164293541075012568708794, 16.93322473697680310058467733670, 17.48878731485655027304264433102

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