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Properties

Label 1-6003-6003.160-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.370 + 0.928i$
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  + (−0.680 − 0.733i)2-s + (−0.0747 + 0.997i)4-s + (0.955 + 0.294i)5-s + (−0.0747 − 0.997i)7-s + (0.781 − 0.623i)8-s + (−0.433 − 0.900i)10-s + (−0.149 + 0.988i)11-s + (−0.365 − 0.930i)13-s + (−0.680 + 0.733i)14-s + (−0.988 − 0.149i)16-s − i·17-s + (0.433 + 0.900i)19-s + (−0.365 + 0.930i)20-s + (0.826 − 0.563i)22-s + (0.826 + 0.563i)25-s + (−0.433 + 0.900i)26-s + ⋯
L(s)  = 1  + (−0.680 − 0.733i)2-s + (−0.0747 + 0.997i)4-s + (0.955 + 0.294i)5-s + (−0.0747 − 0.997i)7-s + (0.781 − 0.623i)8-s + (−0.433 − 0.900i)10-s + (−0.149 + 0.988i)11-s + (−0.365 − 0.930i)13-s + (−0.680 + 0.733i)14-s + (−0.988 − 0.149i)16-s − i·17-s + (0.433 + 0.900i)19-s + (−0.365 + 0.930i)20-s + (0.826 − 0.563i)22-s + (0.826 + 0.563i)25-s + (−0.433 + 0.900i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.370 + 0.928i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.370 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6623747607 + 0.4489722720i\)
\(L(\frac12)\) \(\approx\) \(0.6623747607 + 0.4489722720i\)
\(L(1)\) \(\approx\) \(0.7773595723 - 0.1583622732i\)
\(L(1)\) \(\approx\) \(0.7773595723 - 0.1583622732i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.680 - 0.733i)T \)
5 \( 1 + (0.955 + 0.294i)T \)
7 \( 1 + (-0.0747 - 0.997i)T \)
11 \( 1 + (-0.149 + 0.988i)T \)
13 \( 1 + (-0.365 - 0.930i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.433 + 0.900i)T \)
31 \( 1 + (-0.294 + 0.955i)T \)
37 \( 1 + (-0.781 + 0.623i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.294 - 0.955i)T \)
47 \( 1 + (-0.149 + 0.988i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.997 - 0.0747i)T \)
67 \( 1 + (-0.988 + 0.149i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.974 + 0.222i)T \)
79 \( 1 + (-0.930 - 0.365i)T \)
83 \( 1 + (-0.826 - 0.563i)T \)
89 \( 1 + (-0.974 + 0.222i)T \)
97 \( 1 + (-0.563 + 0.826i)T \)
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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

−17.55048374574018475484857348661, −16.94270570308028888634099564622, −16.338570908075127408415824668622, −15.82318336479359705044176999803, −14.989292323835370692844414114262, −14.46249973249924681173502844922, −13.76201285427406168031632857374, −13.18483388296186472984175616928, −12.40123867998284887559996411686, −11.40396898129633153890567650525, −10.96103447592357879608296591190, −9.95686240492153586447251424336, −9.50752673192460163420080518596, −8.77169723796531078951836589742, −8.540452708839746576981355334706, −7.55183933957420479718705504331, −6.63539186961190845354060799178, −6.166753608432310448586004824266, −5.48900834261601268882818488006, −5.04996952233174375418164897116, −4.02088084541515738902323533391, −2.77774524479258668028490729580, −2.07392199166057717113172275984, −1.437665016046989634364451120041, −0.254559749276812821140560913339, 1.09018823447226913421954067625, 1.57897127715635136961661404460, 2.63375418208882643378365673307, 3.050596305980235031177213045170, 4.00139961907516410475797689020, 4.8172749375387537056918297363, 5.52891157872648737484582041916, 6.639827771016539867393182197178, 7.30638502396559322592590317312, 7.643151433843329751648992558681, 8.64997235468926000296493771905, 9.47786296100258955062895607931, 10.01822603957625018493865558203, 10.339529603407724340988803203115, 10.97576108266536857987269040075, 11.90713999585532151339922110410, 12.57070909882246942852789266675, 13.10428517265141064084880875464, 13.86603313624874161818961914653, 14.30214886013001979983897119441, 15.275571321186380596922318662300, 16.15280624967717591076631647957, 16.72756565767878065768987084249, 17.54291865244943153926461498700, 17.70154354457593989184147387231

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