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Properties

Label 1-6003-6003.1141-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.768 - 0.639i$
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  + (−0.798 + 0.601i)2-s + (0.275 − 0.961i)4-s + (−0.634 + 0.773i)5-s + (−0.534 − 0.844i)7-s + (0.359 + 0.933i)8-s + (0.0407 − 0.999i)10-s + (0.268 − 0.963i)11-s + (−0.994 + 0.108i)13-s + (0.935 + 0.352i)14-s + (−0.848 − 0.529i)16-s + (0.909 − 0.415i)17-s + (−0.242 − 0.970i)19-s + (0.568 + 0.822i)20-s + (0.365 + 0.930i)22-s + (−0.195 − 0.980i)25-s + (0.728 − 0.685i)26-s + ⋯
L(s)  = 1  + (−0.798 + 0.601i)2-s + (0.275 − 0.961i)4-s + (−0.634 + 0.773i)5-s + (−0.534 − 0.844i)7-s + (0.359 + 0.933i)8-s + (0.0407 − 0.999i)10-s + (0.268 − 0.963i)11-s + (−0.994 + 0.108i)13-s + (0.935 + 0.352i)14-s + (−0.848 − 0.529i)16-s + (0.909 − 0.415i)17-s + (−0.242 − 0.970i)19-s + (0.568 + 0.822i)20-s + (0.365 + 0.930i)22-s + (−0.195 − 0.980i)25-s + (0.728 − 0.685i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.768 - 0.639i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.768 - 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1541358000 - 0.4261108955i\)
\(L(\frac12)\) \(\approx\) \(0.1541358000 - 0.4261108955i\)
\(L(1)\) \(\approx\) \(0.5586993287 + 0.009257924801i\)
\(L(1)\) \(\approx\) \(0.5586993287 + 0.009257924801i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.798 + 0.601i)T \)
5 \( 1 + (-0.634 + 0.773i)T \)
7 \( 1 + (-0.534 - 0.844i)T \)
11 \( 1 + (0.268 - 0.963i)T \)
13 \( 1 + (-0.994 + 0.108i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (-0.242 - 0.970i)T \)
31 \( 1 + (0.737 - 0.675i)T \)
37 \( 1 + (-0.998 - 0.0611i)T \)
41 \( 1 + (-0.971 - 0.235i)T \)
43 \( 1 + (0.737 + 0.675i)T \)
47 \( 1 + (-0.294 - 0.955i)T \)
53 \( 1 + (-0.262 - 0.965i)T \)
59 \( 1 + (-0.327 - 0.945i)T \)
61 \( 1 + (0.999 + 0.00679i)T \)
67 \( 1 + (0.963 - 0.268i)T \)
71 \( 1 + (-0.339 - 0.940i)T \)
73 \( 1 + (0.999 - 0.0203i)T \)
79 \( 1 + (-0.999 + 0.0339i)T \)
83 \( 1 + (0.612 - 0.790i)T \)
89 \( 1 + (-0.830 - 0.557i)T \)
97 \( 1 + (-0.996 - 0.0882i)T \)
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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

−18.0136331227610837285280721820, −17.15447561447530417248416946959, −16.94047829954220004451355249647, −16.080066731891543790870037891117, −15.534429724743392122861830667078, −14.921930897522379277135612570001, −14.02579035070784419337393238055, −12.8582030295074554425826038367, −12.39657614219588698661407849017, −12.20532415937699364928032432215, −11.58128822142757295795454162873, −10.48566145268607430526640453452, −9.88574238613674911194912198489, −9.44809528372525311516113049420, −8.60842244678937912915695846509, −8.16296111336978412483655308406, −7.39223959221933647716773195287, −6.77970669616752892684664339362, −5.73115721226410132500837459410, −4.93728898631019394841696725084, −4.13093381431002046339707686818, −3.43313947692734142627147294949, −2.63234078902391813545408167378, −1.80657413268708603502145182980, −1.093458781846248600110655507060, 0.222123048816858598023305776077, 0.76566417665430082507913236794, 2.051749717994044382450448384800, 2.96490481300998983322736920894, 3.57167249463248489412777901398, 4.57441968585853752305745683350, 5.31996806658989438702304594134, 6.32999345706040954397399613332, 6.78197268354860019990417116697, 7.3551890174225107253183021541, 7.96381455943089588206751310518, 8.65741518561720323754881328847, 9.59946355548989167749669692096, 10.0411505940205832454920480413, 10.7097784428392358340081981711, 11.38655744030709321788113834587, 11.87994956074133505429710579365, 12.961347957158794285562802449100, 13.92891099927727618861697225653, 14.19006921788242770141742550223, 14.9595851900787710117993580120, 15.65295855056112508560017513349, 16.16558538192582325937002874353, 16.92070579504735878814914865263, 17.24225511717072171900869762266

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