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Properties

Label 1-6027-6027.3071-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.962 + 0.270i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.420 − 0.907i)2-s + (−0.646 + 0.762i)4-s + (−0.842 + 0.538i)5-s + (0.963 + 0.266i)8-s + (0.842 + 0.538i)10-s + (0.946 − 0.323i)11-s + (0.995 + 0.0896i)13-s + (−0.163 − 0.986i)16-s + (−0.337 + 0.941i)17-s + (−0.669 + 0.743i)19-s + (0.134 − 0.990i)20-s + (−0.691 − 0.722i)22-s + (−0.791 + 0.611i)23-s + (0.420 − 0.907i)25-s + (−0.337 − 0.941i)26-s + ⋯
L(s)  = 1  + (−0.420 − 0.907i)2-s + (−0.646 + 0.762i)4-s + (−0.842 + 0.538i)5-s + (0.963 + 0.266i)8-s + (0.842 + 0.538i)10-s + (0.946 − 0.323i)11-s + (0.995 + 0.0896i)13-s + (−0.163 − 0.986i)16-s + (−0.337 + 0.941i)17-s + (−0.669 + 0.743i)19-s + (0.134 − 0.990i)20-s + (−0.691 − 0.722i)22-s + (−0.791 + 0.611i)23-s + (0.420 − 0.907i)25-s + (−0.337 − 0.941i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.962 + 0.270i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (3071, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.962 + 0.270i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060073438 + 0.1460662759i\)
\(L(\frac12)\) \(\approx\) \(1.060073438 + 0.1460662759i\)
\(L(1)\) \(\approx\) \(0.7506162884 - 0.1353473168i\)
\(L(1)\) \(\approx\) \(0.7506162884 - 0.1353473168i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.420 - 0.907i)T \)
5 \( 1 + (-0.842 + 0.538i)T \)
11 \( 1 + (0.946 - 0.323i)T \)
13 \( 1 + (0.995 + 0.0896i)T \)
17 \( 1 + (-0.337 + 0.941i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (-0.791 + 0.611i)T \)
29 \( 1 + (-0.473 - 0.880i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.525 - 0.850i)T \)
43 \( 1 + (-0.550 - 0.834i)T \)
47 \( 1 + (0.420 + 0.907i)T \)
53 \( 1 + (0.646 - 0.762i)T \)
59 \( 1 + (0.998 - 0.0598i)T \)
61 \( 1 + (0.925 - 0.379i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (-0.473 + 0.880i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.575 + 0.817i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−17.53138324198435460347822777096, −16.92815847606645441708934323983, −16.260391162990163011475465255472, −15.94011673116621528925835431021, −14.98557729752534992854466067813, −14.83057782530119410023738869626, −13.739246057524910951214610703505, −13.27642336728787479811295854396, −12.5068250476512377686883010048, −11.56225702865456469211360314565, −11.19236837557898966022662754407, −10.22274283980300456109495479118, −9.451225001878196892350279017305, −8.78029787554720779206987977786, −8.44835592812549438050247792117, −7.60728455907739825546703475235, −6.920893305445930917820537495726, −6.38745614585016438714709702035, −5.54825372386393007015901422927, −4.66099546880152280264624596367, −4.23629454155118445640179442669, −3.4976947327477040617354113048, −2.201084622582384991938447270630, −1.19003065832673870760064128357, −0.488857588654382703926079852369, 0.74825252354000264784030656602, 1.66493238293350384968299991624, 2.34474467107110245819487298938, 3.48576532215855595030182043767, 3.86886546013108164385094502240, 4.20978125955394969867072546410, 5.54193989250713214651054935821, 6.36647562092567426022471504609, 7.04040044320027096032281766175, 8.04582752474706217298335484751, 8.35834607783616510678577364579, 9.0368136870241506185373753968, 9.93984169566188353357099111535, 10.54721278255929695408050683594, 11.237969696174745341898487008088, 11.57336354376631319497992215413, 12.34807050549922689719019324758, 12.92029948693755825938522792450, 13.80289680920560805507180888585, 14.339746504436063857287860462895, 15.07987254482862588436050507613, 15.94115833811283838808959253543, 16.450885444406763970022892676301, 17.29213559198036236029129002555, 17.7992831895838783153901783387

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