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Properties

Label 1-6001-6001.1124-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.724 - 0.689i$
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  + (0.142 + 0.989i)2-s + (−0.247 − 0.968i)3-s + (−0.959 + 0.281i)4-s + (−0.968 + 0.247i)5-s + (0.923 − 0.382i)6-s + (−0.382 + 0.923i)7-s + (−0.415 − 0.909i)8-s + (−0.877 + 0.479i)9-s + (−0.382 − 0.923i)10-s + (0.0713 + 0.997i)11-s + (0.510 + 0.860i)12-s + (0.984 − 0.177i)13-s + (−0.968 − 0.247i)14-s + (0.479 + 0.877i)15-s + (0.841 − 0.540i)16-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)2-s + (−0.247 − 0.968i)3-s + (−0.959 + 0.281i)4-s + (−0.968 + 0.247i)5-s + (0.923 − 0.382i)6-s + (−0.382 + 0.923i)7-s + (−0.415 − 0.909i)8-s + (−0.877 + 0.479i)9-s + (−0.382 − 0.923i)10-s + (0.0713 + 0.997i)11-s + (0.510 + 0.860i)12-s + (0.984 − 0.177i)13-s + (−0.968 − 0.247i)14-s + (0.479 + 0.877i)15-s + (0.841 − 0.540i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.724 - 0.689i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (1124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.724 - 0.689i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3785623059 - 0.1513941279i\)
\(L(\frac12)\) \(\approx\) \(0.3785623059 - 0.1513941279i\)
\(L(1)\) \(\approx\) \(0.6223161118 + 0.2440628552i\)
\(L(1)\) \(\approx\) \(0.6223161118 + 0.2440628552i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (0.142 + 0.989i)T \)
3 \( 1 + (-0.247 - 0.968i)T \)
5 \( 1 + (-0.968 + 0.247i)T \)
7 \( 1 + (-0.382 + 0.923i)T \)
11 \( 1 + (0.0713 + 0.997i)T \)
13 \( 1 + (0.984 - 0.177i)T \)
19 \( 1 + (-0.212 - 0.977i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.479 - 0.877i)T \)
31 \( 1 + (-0.948 - 0.315i)T \)
37 \( 1 + (0.0356 + 0.999i)T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (-0.997 + 0.0713i)T \)
47 \( 1 + (-0.877 - 0.479i)T \)
53 \( 1 + (0.247 + 0.968i)T \)
59 \( 1 + (0.923 - 0.382i)T \)
61 \( 1 + (-0.212 + 0.977i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (0.106 - 0.994i)T \)
73 \( 1 + (0.877 + 0.479i)T \)
79 \( 1 + (-0.968 - 0.247i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (-0.731 + 0.681i)T \)
97 \( 1 + (0.349 + 0.936i)T \)
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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

−17.962003422196397302348489500234, −16.860633893887471165437607190640, −16.41879113246151607887169614440, −16.14848988234690786222520503513, −15.044749151179421503362879289879, −14.45276822552863013959733514669, −13.92662597418466329329267074347, −12.89582272911331233996906446897, −12.63101429024211694570731017278, −11.47436947034364250888871407207, −11.17594022431137091350985164527, −10.724943815091973157213966876114, −10.038391017770383469310296317691, −9.22448790282791057508173240507, −8.5752124148064949189554824068, −8.15740626609979032906240350976, −6.97619693427629667342608832348, −6.060103709098707133425020100316, −5.34674788904711972813107984230, −4.5263051388806276868285603463, −3.88948097475175894083531958715, −3.50446900828401804973475760527, −2.99423493480597217115563843103, −1.50537508652503406137976897678, −0.71042947058046349599511554108, 0.16021982718976910795218054385, 1.295473613071893690329736737948, 2.411389054927472968651777464865, 3.22526612532759706703402044844, 3.978232738035575182440329326295, 4.94579885182028468634501745073, 5.50989004170752076350486980003, 6.4027208325394671739721055248, 6.76565333927089486552437271575, 7.54687779908967737975576108847, 7.974599014554724969813753002917, 8.8560068451724361340962056011, 9.20148105751830087236227566347, 10.34611054046783385920328042442, 11.36205075481253615511321511129, 11.80019386070094424145915228941, 12.51750033472928806107569442639, 13.17291741019467442995601246433, 13.47414904008328930379502935603, 14.64526086147421156092361517339, 15.11363986938756387355007790213, 15.54416263401818303652928536122, 16.257108309268940694687351570113, 16.99076042461345135193892427770, 17.68306968707941406496462094184

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