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Properties

Label 1-6003-6003.5-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.970 + 0.242i$
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  + (−0.248 − 0.968i)2-s + (−0.876 + 0.482i)4-s + (−0.999 + 0.0135i)5-s + (−0.704 − 0.709i)7-s + (0.685 + 0.728i)8-s + (0.262 + 0.965i)10-s + (0.751 − 0.659i)11-s + (−0.942 + 0.333i)13-s + (−0.511 + 0.859i)14-s + (0.534 − 0.844i)16-s + (−0.415 + 0.909i)17-s + (0.0203 − 0.999i)19-s + (0.869 − 0.494i)20-s + (−0.826 − 0.563i)22-s + (0.999 − 0.0271i)25-s + (0.557 + 0.830i)26-s + ⋯
L(s)  = 1  + (−0.248 − 0.968i)2-s + (−0.876 + 0.482i)4-s + (−0.999 + 0.0135i)5-s + (−0.704 − 0.709i)7-s + (0.685 + 0.728i)8-s + (0.262 + 0.965i)10-s + (0.751 − 0.659i)11-s + (−0.942 + 0.333i)13-s + (−0.511 + 0.859i)14-s + (0.534 − 0.844i)16-s + (−0.415 + 0.909i)17-s + (0.0203 − 0.999i)19-s + (0.869 − 0.494i)20-s + (−0.826 − 0.563i)22-s + (0.999 − 0.0271i)25-s + (0.557 + 0.830i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2861425422 + 0.03515236351i\)
\(L(\frac12)\) \(\approx\) \(0.2861425422 + 0.03515236351i\)
\(L(1)\) \(\approx\) \(0.4883016070 - 0.2895791558i\)
\(L(1)\) \(\approx\) \(0.4883016070 - 0.2895791558i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.248 - 0.968i)T \)
5 \( 1 + (-0.999 + 0.0135i)T \)
7 \( 1 + (-0.704 - 0.709i)T \)
11 \( 1 + (0.751 - 0.659i)T \)
13 \( 1 + (-0.942 + 0.333i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (0.0203 - 0.999i)T \)
31 \( 1 + (-0.155 - 0.987i)T \)
37 \( 1 + (-0.377 + 0.925i)T \)
41 \( 1 + (0.235 + 0.971i)T \)
43 \( 1 + (0.155 - 0.987i)T \)
47 \( 1 + (-0.988 + 0.149i)T \)
53 \( 1 + (-0.591 - 0.806i)T \)
59 \( 1 + (0.327 - 0.945i)T \)
61 \( 1 + (0.976 + 0.215i)T \)
67 \( 1 + (0.751 + 0.659i)T \)
71 \( 1 + (-0.101 - 0.994i)T \)
73 \( 1 + (-0.794 + 0.607i)T \)
79 \( 1 + (0.464 - 0.885i)T \)
83 \( 1 + (-0.634 + 0.773i)T \)
89 \( 1 + (-0.996 - 0.0815i)T \)
97 \( 1 + (-0.951 + 0.307i)T \)
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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

−17.731881540282194571318564489507, −16.84036669671187294920282384886, −16.30086660183688388240468594724, −15.72841978214564588436138483671, −15.25145384458510293318545637709, −14.49075236640712487033374926012, −14.17960051104279171486459754030, −12.95628694572141934577148956592, −12.45609997466803084120876148616, −11.98779814159631115698627440531, −11.065282290288316600054197307580, −10.088870915859618475276804574786, −9.582170089610765983664939379501, −8.91282721648368073303946813605, −8.322657932354736026948021757137, −7.40349917362597609214934979387, −7.09223476025006449556833563689, −6.359862945263958792661524080411, −5.485745517304631471379206545098, −4.85933902045100258412057794581, −4.10878870401690683518801038303, −3.40064183688257769370317476450, −2.454861388006663806283613203571, −1.31076500579797899189768019812, −0.1396625824103983865925580802, 0.607810558128083479163049118974, 1.53432126609370507588975727631, 2.59902916840564857199702057812, 3.27809255043888932327042073897, 3.96348059829747808004867580945, 4.383861217078059489741602017509, 5.24187446761269473690898055568, 6.52310569598171456237147433471, 6.97357565055571296308997626658, 7.89041214206218485741792630794, 8.437566597426009657338980142769, 9.229444610013409488082401850665, 9.791315892891764504970367731, 10.53729770059529020512017910982, 11.33762366492767789834438007125, 11.53958051411148112034964870726, 12.44941039815534730400949200688, 12.99507568938788405309239475943, 13.58921074063697306917813194139, 14.42199313426063236405195990788, 15.002509500394288423018151274501, 15.93965653076951580577217463482, 16.619342529092230680602858612369, 17.122715944581084018326234125707, 17.66912809566890133942768498836

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