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Properties

Label 1-6003-6003.1139-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.544 + 0.838i$
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  + (0.0543 − 0.998i)2-s + (−0.994 − 0.108i)4-s + (−0.912 + 0.409i)5-s + (0.777 − 0.628i)7-s + (−0.162 + 0.986i)8-s + (0.359 + 0.933i)10-s + (−0.346 − 0.938i)11-s + (−0.440 + 0.897i)13-s + (−0.585 − 0.810i)14-s + (0.976 + 0.215i)16-s + (0.755 + 0.654i)17-s + (−0.806 − 0.591i)19-s + (0.951 − 0.307i)20-s + (−0.955 + 0.294i)22-s + (0.665 − 0.746i)25-s + (0.872 + 0.488i)26-s + ⋯
L(s)  = 1  + (0.0543 − 0.998i)2-s + (−0.994 − 0.108i)4-s + (−0.912 + 0.409i)5-s + (0.777 − 0.628i)7-s + (−0.162 + 0.986i)8-s + (0.359 + 0.933i)10-s + (−0.346 − 0.938i)11-s + (−0.440 + 0.897i)13-s + (−0.585 − 0.810i)14-s + (0.976 + 0.215i)16-s + (0.755 + 0.654i)17-s + (−0.806 − 0.591i)19-s + (0.951 − 0.307i)20-s + (−0.955 + 0.294i)22-s + (0.665 − 0.746i)25-s + (0.872 + 0.488i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.544 + 0.838i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.544 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4022483168 + 0.2184913801i\)
\(L(\frac12)\) \(\approx\) \(0.4022483168 + 0.2184913801i\)
\(L(1)\) \(\approx\) \(0.6955501440 - 0.3257274593i\)
\(L(1)\) \(\approx\) \(0.6955501440 - 0.3257274593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.0543 - 0.998i)T \)
5 \( 1 + (-0.912 + 0.409i)T \)
7 \( 1 + (0.777 - 0.628i)T \)
11 \( 1 + (-0.346 - 0.938i)T \)
13 \( 1 + (-0.440 + 0.897i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (-0.806 - 0.591i)T \)
31 \( 1 + (0.135 + 0.990i)T \)
37 \( 1 + (-0.852 + 0.523i)T \)
41 \( 1 + (0.458 - 0.888i)T \)
43 \( 1 + (-0.135 + 0.990i)T \)
47 \( 1 + (0.997 + 0.0747i)T \)
53 \( 1 + (-0.685 - 0.728i)T \)
59 \( 1 + (0.786 - 0.618i)T \)
61 \( 1 + (-0.446 + 0.894i)T \)
67 \( 1 + (-0.938 - 0.346i)T \)
71 \( 1 + (0.0203 + 0.999i)T \)
73 \( 1 + (-0.983 - 0.182i)T \)
79 \( 1 + (0.737 - 0.675i)T \)
83 \( 1 + (-0.644 - 0.764i)T \)
89 \( 1 + (0.574 + 0.818i)T \)
97 \( 1 + (-0.268 - 0.963i)T \)
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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

−17.50881367153000183737665131381, −16.920280650324783311948168426153, −16.24522740890713976115365828047, −15.43603202886949588349972306716, −15.15954267943586295531393248893, −14.64764553380368905553620322805, −13.865698435550881897634645822338, −12.92672237081312950930160050963, −12.315208061887913858116836278001, −12.07104687232641805622257706664, −10.980156406822454875508233743855, −10.17590773741704221859610455886, −9.443962745892751519682838734635, −8.68392603728669624842068716461, −8.07086389428779162808044839707, −7.58294671315384766740235488333, −7.12723371346389616537906069257, −5.918755344754978453511063880610, −5.40543330114833492242834174169, −4.71631870861776979044145940946, −4.23448880968159243058867609159, −3.29618347039649626394431043287, −2.349362994351195052867658923368, −1.22793203763298661423540156044, −0.14505634432598811964881830586, 0.90379225647420251907525455678, 1.716104110194907820532827209540, 2.63874998401497942667053128867, 3.37870169362871731512148005426, 4.02566509609005486357823005428, 4.612410235177679035339893856409, 5.306963692125604299945267991790, 6.34545350519750354553475977934, 7.21291727272033580346718576204, 7.94383490964160594305319798313, 8.51303451427660247012197038478, 9.08908529907135637876194978129, 10.31448298241978020647556032116, 10.54576095121692524396948716356, 11.25034773312253399984466501324, 11.74894450374622187976271614048, 12.366601258571107790363056332389, 13.16382001718386494934977021828, 13.9268625163213223600737198110, 14.43284254379590019805223186664, 14.90801010853421034842015099477, 15.88658106673977292563722265621, 16.63176460843992505133961474683, 17.30211093794122841052337443479, 17.89867150236768892407144037836

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