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Properties

Label 1-6001-6001.43-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.916 + 0.399i$
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  + (−0.959 + 0.281i)2-s + (0.106 + 0.994i)3-s + (0.841 − 0.540i)4-s + (0.994 − 0.106i)5-s + (−0.382 − 0.923i)6-s + (0.923 + 0.382i)7-s + (−0.654 + 0.755i)8-s + (−0.977 + 0.212i)9-s + (−0.923 + 0.382i)10-s + (−0.599 + 0.800i)11-s + (0.627 + 0.778i)12-s + (0.731 − 0.681i)13-s + (−0.994 − 0.106i)14-s + (0.212 + 0.977i)15-s + (0.415 − 0.909i)16-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (0.106 + 0.994i)3-s + (0.841 − 0.540i)4-s + (0.994 − 0.106i)5-s + (−0.382 − 0.923i)6-s + (0.923 + 0.382i)7-s + (−0.654 + 0.755i)8-s + (−0.977 + 0.212i)9-s + (−0.923 + 0.382i)10-s + (−0.599 + 0.800i)11-s + (0.627 + 0.778i)12-s + (0.731 − 0.681i)13-s + (−0.994 − 0.106i)14-s + (0.212 + 0.977i)15-s + (0.415 − 0.909i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.916 + 0.399i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.916 + 0.399i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2771380732 + 1.331370540i\)
\(L(\frac12)\) \(\approx\) \(0.2771380732 + 1.331370540i\)
\(L(1)\) \(\approx\) \(0.7483952325 + 0.5057602553i\)
\(L(1)\) \(\approx\) \(0.7483952325 + 0.5057602553i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (0.106 + 0.994i)T \)
5 \( 1 + (0.994 - 0.106i)T \)
7 \( 1 + (0.923 + 0.382i)T \)
11 \( 1 + (-0.599 + 0.800i)T \)
13 \( 1 + (0.731 - 0.681i)T \)
19 \( 1 + (-0.349 + 0.936i)T \)
23 \( 1 + (-0.989 - 0.142i)T \)
29 \( 1 + (-0.977 + 0.212i)T \)
31 \( 1 + (0.510 + 0.860i)T \)
37 \( 1 + (-0.447 - 0.894i)T \)
41 \( 1 + (0.989 - 0.142i)T \)
43 \( 1 + (-0.599 + 0.800i)T \)
47 \( 1 + (0.977 + 0.212i)T \)
53 \( 1 + (0.994 - 0.106i)T \)
59 \( 1 + (-0.923 + 0.382i)T \)
61 \( 1 + (0.936 - 0.349i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (-0.984 + 0.177i)T \)
73 \( 1 + (0.212 - 0.977i)T \)
79 \( 1 + (0.994 + 0.106i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (0.315 + 0.948i)T \)
97 \( 1 + (-0.0713 + 0.997i)T \)
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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

−17.66598229889571816498546577152, −16.96403234452478589702740614660, −16.5576229063534165192094908832, −15.53046939048951687103382059763, −14.77209314601922016088450395016, −13.84453663997475935164436846936, −13.53289977053225779509885293927, −12.95656985281114313127076901353, −11.89004463521431770675718489089, −11.42918854819628854236244974074, −10.82734298688436514747380770993, −10.2248849432931269511352847115, −9.247311832726538732284288028124, −8.685082656513593566513968699144, −8.175249586824642052400007100343, −7.41777181479052849323491629208, −6.82377284922200425153815662069, −6.0118112728059093016528304398, −5.60743381533030588598976360143, −4.332057553605736598285574640629, −3.32203770171583035290633323275, −2.4266737952544024910118506185, −1.97539393991141626829351538150, −1.27938935816630690058244672117, −0.46253302073308254679133515243, 1.10915896292772460608864384529, 2.03546836236658401810201957161, 2.414459018769738958194418014938, 3.488978973084245976635483252731, 4.5334752620083062182234880554, 5.337051031963157753179455833947, 5.718140230472646813942793082055, 6.37502429730825878360473018609, 7.608677422284676837865383459998, 8.065328239185528661794887050442, 8.81731293659379802381386811014, 9.275635062521378908394639440332, 10.10304864027599978277583246752, 10.53857896192827563082202381681, 10.92076069256432400976146868693, 11.93882703470454067688254072823, 12.5771216522824830699571017321, 13.69484538253047517926634408933, 14.3413974622089673423915364354, 14.909422500402378497814512206759, 15.40398375855943499199690276856, 16.19342347242332219865103789183, 16.628087593371712615020974818913, 17.54074335805695413761680724168, 17.87711759402547721611206690427

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