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Properties

Label 1-6003-6003.608-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.905 - 0.423i$
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  + (−0.786 − 0.618i)2-s + (0.235 + 0.971i)4-s + (−0.327 + 0.945i)5-s + (0.888 − 0.458i)7-s + (0.415 − 0.909i)8-s + (0.841 − 0.540i)10-s + (−0.928 − 0.371i)11-s + (−0.888 − 0.458i)13-s + (−0.981 − 0.189i)14-s + (−0.888 + 0.458i)16-s + (0.959 − 0.281i)17-s + (−0.959 − 0.281i)19-s + (−0.995 − 0.0950i)20-s + (0.5 + 0.866i)22-s + (−0.786 − 0.618i)25-s + (0.415 + 0.909i)26-s + ⋯
L(s)  = 1  + (−0.786 − 0.618i)2-s + (0.235 + 0.971i)4-s + (−0.327 + 0.945i)5-s + (0.888 − 0.458i)7-s + (0.415 − 0.909i)8-s + (0.841 − 0.540i)10-s + (−0.928 − 0.371i)11-s + (−0.888 − 0.458i)13-s + (−0.981 − 0.189i)14-s + (−0.888 + 0.458i)16-s + (0.959 − 0.281i)17-s + (−0.959 − 0.281i)19-s + (−0.995 − 0.0950i)20-s + (0.5 + 0.866i)22-s + (−0.786 − 0.618i)25-s + (0.415 + 0.909i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.905 - 0.423i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (608, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.905 - 0.423i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07792308614 - 0.3504929997i\)
\(L(\frac12)\) \(\approx\) \(0.07792308614 - 0.3504929997i\)
\(L(1)\) \(\approx\) \(0.6094972857 - 0.1241993301i\)
\(L(1)\) \(\approx\) \(0.6094972857 - 0.1241993301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.786 - 0.618i)T \)
5 \( 1 + (-0.327 + 0.945i)T \)
7 \( 1 + (0.888 - 0.458i)T \)
11 \( 1 + (-0.928 - 0.371i)T \)
13 \( 1 + (-0.888 - 0.458i)T \)
17 \( 1 + (0.959 - 0.281i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (0.995 - 0.0950i)T \)
37 \( 1 + (-0.654 + 0.755i)T \)
41 \( 1 + (-0.327 + 0.945i)T \)
43 \( 1 + (-0.995 - 0.0950i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (0.888 + 0.458i)T \)
61 \( 1 + (0.580 + 0.814i)T \)
67 \( 1 + (-0.928 + 0.371i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (0.0475 + 0.998i)T \)
83 \( 1 + (-0.327 - 0.945i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (0.981 - 0.189i)T \)
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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

−17.83621434837617796341515510194, −17.26103014707702742845176900036, −16.807109919862837525999867463527, −16.10892419609539432231112177366, −15.43433337031651586245055694860, −14.909204950829804011048404254635, −14.34788898268013832213117151046, −13.500435059202555104279592858726, −12.59041689993638144659661531296, −12.021785952519972098231330152865, −11.430068100882103217799973610428, −10.4349992563288387745226669897, −10.01930403299253632629667604019, −9.17737465889463002329783120494, −8.43672107054267244166517560526, −8.12574350372284198014524111755, −7.46761741233121722075100405991, −6.70855364896729809774226739000, −5.67445176607829241369969783821, −5.10755650131572486457956731985, −4.75843541496985178188055698369, −3.728827833829170838860959137485, −2.251274049796218248177806258926, −1.95277492185114460911716492326, −0.925666470761190794418972448056, 0.14149606234560591397258846702, 1.15450580889418108179927060466, 2.158639620303577710355627018890, 2.816689300676492846933144426711, 3.33971099408522952075550956939, 4.34669019580817623069423579302, 4.97580835080630898351091422, 6.04710268297168999054756086363, 7.03252806372370079947417125620, 7.446053872889786284593724431794, 8.194947858878192480381804658998, 8.46122032101562017444706413586, 9.80084761232038422928230843329, 10.25129878762032585487723377040, 10.63335688782750909500979283920, 11.49219238999165728452408612839, 11.80487270852007097379336794826, 12.70370700204131749959694590215, 13.44880010081668680656631596756, 14.125705900080176110015827674352, 14.98814614883448292080103058092, 15.36489622293710838452317248607, 16.3303674010930296112807843392, 16.98160721157262626785589167412, 17.54460792026753125528537058077

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