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Properties

Label 1-6003-6003.1757-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.365 + 0.930i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
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  + (0.458 − 0.888i)2-s + (−0.580 − 0.814i)4-s + (0.235 − 0.971i)5-s + (−0.327 + 0.945i)7-s + (−0.989 + 0.142i)8-s + (−0.755 − 0.654i)10-s + (0.998 + 0.0475i)11-s + (0.327 + 0.945i)13-s + (0.690 + 0.723i)14-s + (−0.327 + 0.945i)16-s + (−0.909 + 0.415i)17-s + (−0.909 − 0.415i)19-s + (−0.928 + 0.371i)20-s + (0.5 − 0.866i)22-s + (−0.888 − 0.458i)25-s + (0.989 + 0.142i)26-s + ⋯
L(s)  = 1  + (0.458 − 0.888i)2-s + (−0.580 − 0.814i)4-s + (0.235 − 0.971i)5-s + (−0.327 + 0.945i)7-s + (−0.989 + 0.142i)8-s + (−0.755 − 0.654i)10-s + (0.998 + 0.0475i)11-s + (0.327 + 0.945i)13-s + (0.690 + 0.723i)14-s + (−0.327 + 0.945i)16-s + (−0.909 + 0.415i)17-s + (−0.909 − 0.415i)19-s + (−0.928 + 0.371i)20-s + (0.5 − 0.866i)22-s + (−0.888 − 0.458i)25-s + (0.989 + 0.142i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.365 + 0.930i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.365 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1835805484 - 0.2691886600i\)
\(L(\frac12)\) \(\approx\) \(-0.1835805484 - 0.2691886600i\)
\(L(1)\) \(\approx\) \(0.8652129437 - 0.5735154220i\)
\(L(1)\) \(\approx\) \(0.8652129437 - 0.5735154220i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.458 - 0.888i)T \)
5 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (-0.327 + 0.945i)T \)
11 \( 1 + (0.998 + 0.0475i)T \)
13 \( 1 + (0.327 + 0.945i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (-0.909 - 0.415i)T \)
31 \( 1 + (0.371 - 0.928i)T \)
37 \( 1 + (0.281 - 0.959i)T \)
41 \( 1 + (-0.971 - 0.235i)T \)
43 \( 1 + (-0.371 - 0.928i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (0.618 + 0.786i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (0.189 - 0.981i)T \)
83 \( 1 + (-0.235 - 0.971i)T \)
89 \( 1 + (-0.989 - 0.142i)T \)
97 \( 1 + (-0.690 + 0.723i)T \)
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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

−17.86232382035941595059707612311, −17.52101896378232937614649121004, −16.784770478557202648691713728731, −16.26118933825967792615893985191, −15.30756959842272759525769103987, −15.01088751837064860738092329044, −14.17366406243827927646929195590, −13.78728594250079136915608320602, −13.1151426648764345762022630618, −12.52816880950876986478506786622, −11.46467383170081914584899082994, −11.02924420235595535376336762009, −9.98312901941660921470068261688, −9.713948179182287486037974761121, −8.46255951172481245543579005904, −8.16824975735289167451426650257, −7.0536927517616255086247297816, −6.66930337112177015088724287764, −6.34262979075841783982689982493, −5.344100550997325020488953275929, −4.530736833386931162038162957340, −3.71759524788916960073086220410, −3.33410223990830332991079473601, −2.43234362054441115074238574064, −1.21872193936876566829093224564, 0.069379444080380706157983484881, 1.233972287338268435303465987150, 2.0157968184509849050024501749, 2.39192859189638647243285386971, 3.65319954444511618104656247875, 4.20096517591961117890818429293, 4.77133035969874283665578681038, 5.6650572344409491434413307669, 6.23757690456701015398530709436, 6.815496803949947121852733324618, 8.3522608213216488301573689, 8.82534068343109353443960007693, 9.24776403949606326540519304181, 9.82519996918411063938086840729, 10.827162120615350928242872848413, 11.490602105527905679003280052711, 12.08286668485068447407774901588, 12.50919180802652346682214151661, 13.38353652034275971756962329367, 13.598120748461458567154184720889, 14.65588167828238069137944680671, 15.14283533212089562403004575758, 15.85959481445926780570718149786, 16.69419368398784265044234940898, 17.29415278789088995080232456671

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