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Properties

Label 1-6003-6003.5515-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.911 - 0.412i$
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  + (−0.963 − 0.268i)2-s + (0.855 + 0.517i)4-s + (0.511 + 0.859i)5-s + (−0.966 + 0.255i)7-s + (−0.685 − 0.728i)8-s + (−0.262 − 0.965i)10-s + (0.195 + 0.980i)11-s + (0.760 + 0.649i)13-s + (0.999 + 0.0135i)14-s + (0.464 + 0.885i)16-s + (−0.415 + 0.909i)17-s + (−0.0203 + 0.999i)19-s + (−0.00679 + 0.999i)20-s + (0.0747 − 0.997i)22-s + (−0.476 + 0.879i)25-s + (−0.557 − 0.830i)26-s + ⋯
L(s)  = 1  + (−0.963 − 0.268i)2-s + (0.855 + 0.517i)4-s + (0.511 + 0.859i)5-s + (−0.966 + 0.255i)7-s + (−0.685 − 0.728i)8-s + (−0.262 − 0.965i)10-s + (0.195 + 0.980i)11-s + (0.760 + 0.649i)13-s + (0.999 + 0.0135i)14-s + (0.464 + 0.885i)16-s + (−0.415 + 0.909i)17-s + (−0.0203 + 0.999i)19-s + (−0.00679 + 0.999i)20-s + (0.0747 − 0.997i)22-s + (−0.476 + 0.879i)25-s + (−0.557 − 0.830i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.911 - 0.412i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (5515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.911 - 0.412i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1254447748 + 0.5818906252i\)
\(L(\frac12)\) \(\approx\) \(-0.1254447748 + 0.5818906252i\)
\(L(1)\) \(\approx\) \(0.5873745151 + 0.2542499154i\)
\(L(1)\) \(\approx\) \(0.5873745151 + 0.2542499154i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.963 - 0.268i)T \)
5 \( 1 + (0.511 + 0.859i)T \)
7 \( 1 + (-0.966 + 0.255i)T \)
11 \( 1 + (0.195 + 0.980i)T \)
13 \( 1 + (0.760 + 0.649i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (-0.0203 + 0.999i)T \)
31 \( 1 + (-0.777 + 0.628i)T \)
37 \( 1 + (0.377 - 0.925i)T \)
41 \( 1 + (-0.723 + 0.690i)T \)
43 \( 1 + (-0.777 - 0.628i)T \)
47 \( 1 + (-0.365 + 0.930i)T \)
53 \( 1 + (-0.591 - 0.806i)T \)
59 \( 1 + (0.981 - 0.189i)T \)
61 \( 1 + (0.675 - 0.737i)T \)
67 \( 1 + (-0.195 + 0.980i)T \)
71 \( 1 + (0.101 + 0.994i)T \)
73 \( 1 + (-0.794 + 0.607i)T \)
79 \( 1 + (-0.534 - 0.844i)T \)
83 \( 1 + (-0.352 - 0.935i)T \)
89 \( 1 + (-0.996 - 0.0815i)T \)
97 \( 1 + (-0.209 + 0.977i)T \)
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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

−17.21153221948917086245260093378, −16.64792411720623820578596364853, −16.27015806141818264305475595623, −15.62748270137954203494785835289, −15.05026192001398826400754824855, −13.80913396664860010530049759461, −13.51416721102519312393636321512, −12.85234341469340790824461329105, −11.91758654512165682771564697034, −11.23429232315935639159454509701, −10.6266775824042779799596096894, −9.73364542938687921852354195476, −9.41863949819403779375547301979, −8.58800283046683144103068718245, −8.31012131573366049624055194909, −7.1741939769061017744015711819, −6.63925258280101818392744680496, −5.86519552993198977642001981497, −5.43167362191470595309350878983, −4.42196038964163468845734604810, −3.29113063347152334365435387208, −2.7517708128601031338274825221, −1.69782226622228548932934874770, −0.81567239854321433508375609477, −0.24801356651749748896262835040, 1.46487655849525809964809669801, 1.905987600615653593609442648582, 2.70100620286196240684535057032, 3.58339793013486908005753181832, 3.99359708353860217690586172022, 5.48664145246447339027717874974, 6.27173288676809217505623295333, 6.674682874893295493716564517003, 7.23703059260068967824238338712, 8.18041398387992965457506897198, 8.90378018106779271336245142537, 9.564262200966607728333349974985, 10.08166647942642779502301761282, 10.58698479078033451852036098349, 11.37090631051442586325775879843, 11.99874643129739664301101052388, 12.91290230970074920870812711087, 13.14191413252898217476528233262, 14.49107023564773289643706881411, 14.71672980370037980150218808377, 15.78002867051431453670579762597, 16.07796722901861719309221211539, 16.97563738591171585303745156814, 17.50233665546973057166702283371, 18.18234700059147651926369800129

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