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Properties

Label 1-6003-6003.4864-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.829 - 0.558i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (4864, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.829 - 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5532689127 - 0.1687178791i\)
\(L(\frac12)\) \(\approx\) \(0.5532689127 - 0.1687178791i\)
\(L(1)\) \(\approx\) \(0.6086786653 - 0.1716121971i\)
\(L(1)\) \(\approx\) \(0.6086786653 - 0.1716121971i\)

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

−17.312263151778563852259872228232, −17.109289918934052810041111173033, −16.441658111036887384805320045711, −15.71444725388877291431135520253, −15.4595718441961869756499905784, −14.568398667172219487882796958553, −13.94309144856928170131144269187, −13.23348158413732707311612411060, −12.50418414139947159064503092127, −12.30922850515077442223287444525, −10.873211853540190656513103790779, −10.6301767666747306125045411420, −9.56163131362370026095174306826, −8.9268643802116999047332991408, −8.21691438234866967990773680069, −7.83641432942033930579857887977, −6.885945879631095174619610081160, −6.47455653678577431814652265123, −5.615440230680539365432072845944, −4.92088757598648626881950149056, −3.88634011489774513193491786418, −3.79178552226249468931307406421, −2.833034372993551937949579239219, −1.279290343426271664687265741133, −0.39520905917795762634289355088, 0.37205173530389411401209426533, 1.77856239592051840522233112092, 2.44307180568335719702915771101, 2.99151901420573057956616129347, 4.00103631076265255587322342183, 4.38077194107459009561499651591, 5.10847167186925606539906529103, 6.25058166588465185175602286359, 6.876718272125587410225894336799, 7.56949728473798629029992875914, 8.583131864144185953602606335666, 9.163787847116824558022898526912, 9.631640869874542730984184448668, 10.457978297888602149325323176922, 11.19381262073102369715749159384, 11.748739028734701015601763203798, 12.279491970438799851442686058734, 12.801716351324460346923480763574, 13.59985464763808835230128735077, 14.33864423928144083550949843127, 15.106165353750199781319551524127, 15.44192617476765018568633025003, 16.362855870012850996001643264477, 17.13692654177881141831154688518, 17.837303477888423498215832239114

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