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Properties

Label 1-6027-6027.158-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.345 + 0.938i$
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  + (0.850 − 0.525i)2-s + (0.447 − 0.894i)4-s + (−0.486 − 0.873i)5-s + (−0.0896 − 0.995i)8-s + (−0.873 − 0.486i)10-s + (0.480 + 0.877i)11-s + (−0.287 − 0.957i)13-s + (−0.599 − 0.800i)16-s + (−0.663 − 0.748i)17-s + (0.933 + 0.358i)19-s + (−0.998 + 0.0448i)20-s + (0.869 + 0.493i)22-s + (−0.842 − 0.538i)23-s + (−0.525 + 0.850i)25-s + (−0.748 − 0.663i)26-s + ⋯
L(s)  = 1  + (0.850 − 0.525i)2-s + (0.447 − 0.894i)4-s + (−0.486 − 0.873i)5-s + (−0.0896 − 0.995i)8-s + (−0.873 − 0.486i)10-s + (0.480 + 0.877i)11-s + (−0.287 − 0.957i)13-s + (−0.599 − 0.800i)16-s + (−0.663 − 0.748i)17-s + (0.933 + 0.358i)19-s + (−0.998 + 0.0448i)20-s + (0.869 + 0.493i)22-s + (−0.842 − 0.538i)23-s + (−0.525 + 0.850i)25-s + (−0.748 − 0.663i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.345 + 0.938i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.345 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2480230189 + 0.1729310645i\)
\(L(\frac12)\) \(\approx\) \(0.2480230189 + 0.1729310645i\)
\(L(1)\) \(\approx\) \(1.094267639 - 0.6172274077i\)
\(L(1)\) \(\approx\) \(1.094267639 - 0.6172274077i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.850 - 0.525i)T \)
5 \( 1 + (-0.486 - 0.873i)T \)
11 \( 1 + (0.480 + 0.877i)T \)
13 \( 1 + (-0.287 - 0.957i)T \)
17 \( 1 + (-0.663 - 0.748i)T \)
19 \( 1 + (0.933 + 0.358i)T \)
23 \( 1 + (-0.842 - 0.538i)T \)
29 \( 1 + (-0.413 + 0.910i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.163 + 0.986i)T \)
43 \( 1 + (-0.657 + 0.753i)T \)
47 \( 1 + (0.229 - 0.973i)T \)
53 \( 1 + (0.316 + 0.948i)T \)
59 \( 1 + (-0.193 + 0.981i)T \)
61 \( 1 + (-0.460 + 0.887i)T \)
67 \( 1 + (-0.838 - 0.544i)T \)
71 \( 1 + (-0.910 + 0.413i)T \)
73 \( 1 + (-0.997 + 0.0747i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.727 - 0.685i)T \)
97 \( 1 + (-0.891 + 0.453i)T \)
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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

−17.46092049087019468673437833389, −16.78135157650609895269938678351, −16.05593739889925290302917848030, −15.630849693900522531583337391072, −14.85992917241544512506118191497, −14.336893814840032242842492969628, −13.77699905167030495694312391595, −13.23082604230180056427879401204, −12.26210140732391940503341147997, −11.50097537499954781736730247763, −11.4012681323591643720001011396, −10.52726260504956890576233856400, −9.477556131159033192440059701966, −8.798768003877842314391846381011, −7.89769828403769644355391550777, −7.43784425986361694094125161351, −6.66541467029808323682607590249, −6.144540633533464385574266238519, −5.50227916081590183477010235829, −4.44005224755593119015044084498, −3.84520585710563447396596305818, −3.39953744796985948797923360814, −2.4228149933461509486234807084, −1.77190109585544964042169131047, −0.051830004673295831531059918772, 1.14344209982172993893819638601, 1.70245296597362040632237522192, 2.751045848055443058872711819276, 3.42631669448841032324898115933, 4.24970699419717188491384911942, 4.82100108063814941177476622346, 5.35249166902712199728391055421, 6.14253623583509314352544178575, 7.16490157154281948567007852061, 7.52631771541044998384071737361, 8.641743049859892131584146338, 9.26679455638038043904150136251, 10.017845290116386440825481428456, 10.57916366706153571126163087678, 11.532888772629576775152852203131, 12.138722866721423431336304670444, 12.340503641052850601888604210927, 13.286457082199296760445320313361, 13.63295235272589741160385391838, 14.76242990936385423079464996463, 14.93626056408461268783674778390, 15.93269484121665358903134528475, 16.21526903231535361041905678535, 17.08503110269947881259720273644, 18.06802822883850127338066039542

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