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Properties

Label 1-6001-6001.1133-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.915 - 0.401i$
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  + (−0.755 + 0.654i)2-s + (0.264 + 0.964i)3-s + (0.142 − 0.989i)4-s + (0.868 + 0.494i)5-s + (−0.831 − 0.555i)6-s + (0.980 + 0.195i)7-s + (0.540 + 0.841i)8-s + (−0.860 + 0.510i)9-s + (−0.980 + 0.195i)10-s + (−0.0356 + 0.999i)11-s + (0.992 − 0.124i)12-s + (0.298 + 0.954i)13-s + (−0.868 + 0.494i)14-s + (−0.247 + 0.968i)15-s + (−0.959 − 0.281i)16-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)2-s + (0.264 + 0.964i)3-s + (0.142 − 0.989i)4-s + (0.868 + 0.494i)5-s + (−0.831 − 0.555i)6-s + (0.980 + 0.195i)7-s + (0.540 + 0.841i)8-s + (−0.860 + 0.510i)9-s + (−0.980 + 0.195i)10-s + (−0.0356 + 0.999i)11-s + (0.992 − 0.124i)12-s + (0.298 + 0.954i)13-s + (−0.868 + 0.494i)14-s + (−0.247 + 0.968i)15-s + (−0.959 − 0.281i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.915 - 0.401i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.915 - 0.401i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3547724185 + 1.691659222i\)
\(L(\frac12)\) \(\approx\) \(-0.3547724185 + 1.691659222i\)
\(L(1)\) \(\approx\) \(0.6607884047 + 0.8237162295i\)
\(L(1)\) \(\approx\) \(0.6607884047 + 0.8237162295i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.755 + 0.654i)T \)
3 \( 1 + (0.264 + 0.964i)T \)
5 \( 1 + (0.868 + 0.494i)T \)
7 \( 1 + (0.980 + 0.195i)T \)
11 \( 1 + (-0.0356 + 0.999i)T \)
13 \( 1 + (0.298 + 0.954i)T \)
19 \( 1 + (0.994 + 0.106i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (0.247 - 0.968i)T \)
31 \( 1 + (-0.973 - 0.229i)T \)
37 \( 1 + (0.399 - 0.916i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.681 + 0.731i)T \)
47 \( 1 + (-0.860 - 0.510i)T \)
53 \( 1 + (-0.264 - 0.964i)T \)
59 \( 1 + (-0.555 + 0.831i)T \)
61 \( 1 + (-0.106 - 0.994i)T \)
67 \( 1 + (0.831 - 0.555i)T \)
71 \( 1 + (0.332 - 0.943i)T \)
73 \( 1 + (-0.860 - 0.510i)T \)
79 \( 1 + (-0.868 + 0.494i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (-0.999 + 0.0178i)T \)
97 \( 1 + (-0.177 + 0.984i)T \)
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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

−17.53440915145955527564232729627, −16.980841289158850998170976381946, −16.388228872926033273528000863895, −15.49095179215778643768063507228, −14.32301766584398423768596228564, −13.98944782998976038935126867244, −13.24589093635540284890435238918, −12.7228766883170045624415094892, −12.135803716553472944956679913909, −11.24617186197037119274503829427, −10.85606928128314626926487057666, −10.06302159002915991582131579491, −9.16748185607153775071259596395, −8.46869108726303694184794094445, −8.349602585911463779904435171420, −7.39309776614212308997176697058, −6.8079923806935115896058488735, −5.7497576729670815342637232307, −5.28692656201451202738868769708, −4.18004411671474587403950531562, −3.028123085102097802337095272676, −2.76798169975680573700008851276, −1.598438448051573918461562042518, −1.27500430844722399828167728106, −0.52571473478760009817330994763, 1.37793461705776152255060010113, 1.941759694911021178418305011044, 2.64732835476818857162002081019, 3.80272355660458186356984705258, 4.72720256202099491712344002620, 5.200429756246374324444689911159, 5.8768800216351853097535823290, 6.66339407288672127755931331440, 7.54291083857020438118415234824, 7.97686386702003362316474117002, 9.05493810665515572021403294467, 9.38594771185383709305071037580, 9.87291843896751170161366698773, 10.6231511037566551773494509534, 11.33492568591388090921053880267, 11.66128465955888528162213635940, 13.14235908884012710270617542902, 13.907014115245081730909633901874, 14.41690688408113046760681972980, 14.82586761642477631727231148657, 15.463996619076987817041314993925, 16.11827782293366679724984093825, 16.889936354715984868104778628593, 17.3592502352267067668244510759, 18.05017014545415090286960236445

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