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Properties

Label 1-6003-6003.2-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.740 - 0.672i$
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  + (−0.746 + 0.665i)2-s + (0.115 − 0.993i)4-s + (−0.248 − 0.968i)5-s + (0.440 − 0.897i)7-s + (0.574 + 0.818i)8-s + (0.830 + 0.557i)10-s + (0.719 − 0.694i)11-s + (0.00679 + 0.999i)13-s + (0.268 + 0.963i)14-s + (−0.973 − 0.229i)16-s + (0.755 − 0.654i)17-s + (−0.396 − 0.917i)19-s + (−0.990 + 0.135i)20-s + (−0.0747 + 0.997i)22-s + (−0.876 + 0.482i)25-s + (−0.670 − 0.742i)26-s + ⋯
L(s)  = 1  + (−0.746 + 0.665i)2-s + (0.115 − 0.993i)4-s + (−0.248 − 0.968i)5-s + (0.440 − 0.897i)7-s + (0.574 + 0.818i)8-s + (0.830 + 0.557i)10-s + (0.719 − 0.694i)11-s + (0.00679 + 0.999i)13-s + (0.268 + 0.963i)14-s + (−0.973 − 0.229i)16-s + (0.755 − 0.654i)17-s + (−0.396 − 0.917i)19-s + (−0.990 + 0.135i)20-s + (−0.0747 + 0.997i)22-s + (−0.876 + 0.482i)25-s + (−0.670 − 0.742i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.740 - 0.672i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.740 - 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3137110083 - 0.8121396631i\)
\(L(\frac12)\) \(\approx\) \(0.3137110083 - 0.8121396631i\)
\(L(1)\) \(\approx\) \(0.7337392804 - 0.1465798958i\)
\(L(1)\) \(\approx\) \(0.7337392804 - 0.1465798958i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.746 + 0.665i)T \)
5 \( 1 + (-0.248 - 0.968i)T \)
7 \( 1 + (0.440 - 0.897i)T \)
11 \( 1 + (0.719 - 0.694i)T \)
13 \( 1 + (0.00679 + 0.999i)T \)
17 \( 1 + (0.755 - 0.654i)T \)
19 \( 1 + (-0.396 - 0.917i)T \)
31 \( 1 + (0.859 - 0.511i)T \)
37 \( 1 + (0.994 + 0.101i)T \)
41 \( 1 + (0.458 + 0.888i)T \)
43 \( 1 + (-0.859 - 0.511i)T \)
47 \( 1 + (-0.930 - 0.365i)T \)
53 \( 1 + (-0.996 - 0.0815i)T \)
59 \( 1 + (0.786 + 0.618i)T \)
61 \( 1 + (-0.773 + 0.634i)T \)
67 \( 1 + (0.694 - 0.719i)T \)
71 \( 1 + (-0.452 + 0.891i)T \)
73 \( 1 + (-0.470 + 0.882i)T \)
79 \( 1 + (-0.957 - 0.288i)T \)
83 \( 1 + (-0.601 - 0.798i)T \)
89 \( 1 + (0.998 - 0.0611i)T \)
97 \( 1 + (-0.879 - 0.476i)T \)
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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

−17.94297847615198751332057378336, −17.61582443858068383139062352639, −16.90185460981148641449099439756, −16.00342875222709634015033513723, −15.34331454505683537986358780907, −14.69967372871586994080190993743, −14.27125598610480237592992321849, −13.08786037949912470480231610820, −12.39199934262629591418915513207, −12.02999826502658834200349247167, −11.28166671465877791832657741427, −10.677664401079443587744473660559, −9.987777537702227105475233374970, −9.55657518114924198083885363494, −8.51288833709097110127464986191, −8.022445715603359838643936052914, −7.52160112427455500043241238869, −6.51561396167990048758737225755, −6.01720844395850582217352362750, −4.91782047449824154793937681660, −4.00333787411406784664803797498, −3.32485020784062808252115933215, −2.67726347979017293855992523311, −1.89237948706118147141582943631, −1.21339441941996780072758500669, 0.31451032462734724079903999896, 1.12468476753653199520839334429, 1.6072531766866656127873678682, 2.83968435195053306335657766401, 4.08645448403682883666098070439, 4.518484374426387501638133065841, 5.22183123720475568150814178766, 6.113209232118908694146955709374, 6.79548901408021119018662264182, 7.4343983331171276402597573002, 8.21925569074756185317540678567, 8.61530097066581205149070108693, 9.4883306298056273352851385915, 9.796263617559851481852260683877, 10.86894644253795846352467215576, 11.54568920229339783765001519870, 11.81155635902257859941831442870, 13.18878511258431755905852604759, 13.57037860975196717236824669590, 14.35250300358654566252661811191, 14.80248779629131127139838437845, 15.81981540674066778584843818715, 16.32520771688960942729334982032, 16.86650632243471773665583715502, 17.17109742862021407583582836590

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