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Properties

Label 1-6001-6001.2988-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.00259 + 0.999i$
Analytic cond. $27.8685$
Root an. cond. $27.8685$
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  + (0.989 + 0.142i)2-s + (0.968 + 0.247i)3-s + (0.959 + 0.281i)4-s + (−0.247 + 0.968i)5-s + (0.923 + 0.382i)6-s + (0.923 − 0.382i)7-s + (0.909 + 0.415i)8-s + (0.877 + 0.479i)9-s + (−0.382 + 0.923i)10-s + (0.654 + 0.755i)11-s + (0.860 + 0.510i)12-s + (0.570 + 0.821i)13-s + (0.968 − 0.247i)14-s + (−0.479 + 0.877i)15-s + (0.841 + 0.540i)16-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)2-s + (0.968 + 0.247i)3-s + (0.959 + 0.281i)4-s + (−0.247 + 0.968i)5-s + (0.923 + 0.382i)6-s + (0.923 − 0.382i)7-s + (0.909 + 0.415i)8-s + (0.877 + 0.479i)9-s + (−0.382 + 0.923i)10-s + (0.654 + 0.755i)11-s + (0.860 + 0.510i)12-s + (0.570 + 0.821i)13-s + (0.968 − 0.247i)14-s + (−0.479 + 0.877i)15-s + (0.841 + 0.540i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00259 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00259 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.00259 + 0.999i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (2988, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.00259 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.697373554 + 4.709587849i\)
\(L(\frac12)\) \(\approx\) \(4.697373554 + 4.709587849i\)
\(L(1)\) \(\approx\) \(2.821996698 + 1.313179521i\)
\(L(1)\) \(\approx\) \(2.821996698 + 1.313179521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (0.989 + 0.142i)T \)
3 \( 1 + (0.968 + 0.247i)T \)
5 \( 1 + (-0.247 + 0.968i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
13 \( 1 + (0.570 + 0.821i)T \)
19 \( 1 + (0.212 - 0.977i)T \)
23 \( 1 + (-0.997 + 0.0713i)T \)
29 \( 1 + (-0.959 + 0.281i)T \)
31 \( 1 + (-0.315 - 0.948i)T \)
37 \( 1 + (0.0356 - 0.999i)T \)
41 \( 1 + (-0.0713 + 0.997i)T \)
43 \( 1 + (-0.997 - 0.0713i)T \)
47 \( 1 + (0.479 + 0.877i)T \)
53 \( 1 + (-0.860 + 0.510i)T \)
59 \( 1 + (0.382 + 0.923i)T \)
61 \( 1 + (-0.841 - 0.540i)T \)
67 \( 1 + (0.923 + 0.382i)T \)
71 \( 1 + (-0.106 - 0.994i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (0.247 + 0.968i)T \)
83 \( 1 + (-0.909 + 0.415i)T \)
89 \( 1 + (-0.0356 - 0.999i)T \)
97 \( 1 + (-0.909 + 0.415i)T \)
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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

−17.48454100677889404968663576275, −16.68324440727655549656362909104, −15.98834054975161711970081375568, −15.460496117683328756796881913566, −14.798699190641696173278314533799, −14.18034201225342685372895623140, −13.669767568823143349506758010184, −13.06260488012973550737095424577, −12.28287845230273183297681022656, −11.941514660153951576557971985935, −11.17271751158762447548137522348, −10.321671025725030295151997586758, −9.51320490567801082537366486809, −8.59632725182327544492999414652, −8.18286190386483846077047324564, −7.67880967894197532662095845293, −6.665605740224195233296736507914, −5.80386044187586643716434364086, −5.29443851033368044160003507174, −4.44850703607990568595380134448, −3.62794256976460131246235856083, −3.41248451238709227937503478344, −2.13890339405144267497889007630, −1.61224335610667328054425123965, −0.95547661991137806513562549537, 1.53831152875454471467512891689, 1.979638063792645419491617928192, 2.75417329644679752450388865084, 3.62733856656856492188706615383, 4.180787563642869827016945723915, 4.53345738761761603974646021882, 5.61662767923505843159021540144, 6.567686684513318894510960853976, 7.12560696998840256070492694101, 7.64511079721266687648759261492, 8.26644642687804396042210195817, 9.28959904701859809652512645252, 9.91686475313406194108733701814, 10.978696657504645988701364395446, 11.15254046331814037332395821582, 11.93288031055649399491768806200, 12.787796503257051587989786471209, 13.673772422413361228578459620274, 14.003684786710850310002741699123, 14.56856748718528014544405621778, 15.07813986426683027118693065197, 15.54459064381015742681157412232, 16.35453658679986973360355054607, 17.06695596689510908162918684897, 17.98011101703282529009970661028

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