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Properties

Label 1-6001-6001.83-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.931 + 0.363i$
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 + (0.142 + 0.989i)3-s + (0.959 + 0.281i)4-s + (0.142 + 0.989i)5-s + i·6-s − 7-s + (0.909 + 0.415i)8-s + (−0.959 + 0.281i)9-s + i·10-s + (0.0713 − 0.997i)11-s + (−0.142 + 0.989i)12-s + (−0.212 − 0.977i)13-s + (−0.989 − 0.142i)14-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)2-s + (0.142 + 0.989i)3-s + (0.959 + 0.281i)4-s + (0.142 + 0.989i)5-s + i·6-s − 7-s + (0.909 + 0.415i)8-s + (−0.959 + 0.281i)9-s + i·10-s + (0.0713 − 0.997i)11-s + (−0.142 + 0.989i)12-s + (−0.212 − 0.977i)13-s + (−0.989 − 0.142i)14-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.931 + 0.363i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.931 + 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.894820165 + 0.5443238085i\)
\(L(\frac12)\) \(\approx\) \(2.894820165 + 0.5443238085i\)
\(L(1)\) \(\approx\) \(1.669386025 + 0.6598718519i\)
\(L(1)\) \(\approx\) \(1.669386025 + 0.6598718519i\)

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 \( 1 + (0.142 + 0.989i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 - T \)
11 \( 1 + (0.0713 - 0.997i)T \)
13 \( 1 + (-0.212 - 0.977i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-0.0713 - 0.997i)T \)
29 \( 1 + (0.479 - 0.877i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (-0.909 - 0.415i)T \)
41 \( 1 + (-0.997 - 0.0713i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (-0.281 + 0.959i)T \)
53 \( 1 + (-0.599 + 0.800i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.212 - 0.977i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (0.281 - 0.959i)T \)
73 \( 1 + (0.877 - 0.479i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (0.909 - 0.415i)T \)
89 \( 1 + (0.349 - 0.936i)T \)
97 \( 1 + (0.936 + 0.349i)T \)
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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

−17.570140386071156682755967937852, −16.92561500249627234716426079050, −16.27735323041578304750037987557, −15.75665985627918268548319593333, −14.91323215137835592351075867636, −14.03376006949354906232810060182, −13.75112489568953231627569839106, −12.95742171007194510898981897490, −12.53185441089798375610684789834, −11.98494744448170706092888674320, −11.61283811066821783212141026933, −10.378292595520719341904449936846, −9.668937820307326367689378499788, −9.081237954117183527892116187, −8.18721926116747654220956161471, −7.18958470778020337722298844120, −6.95513897014408864956802704752, −6.18721490558501810781251301169, −5.2915988217488736933455999913, −4.97205235094037149446908477110, −3.72365176649263882790330344374, −3.40688227227148650797346753857, −2.187284004925610836911755056253, −1.76966766877558301758610403957, −0.99065219424874084050210988024, 0.50912266096654878468435286718, 2.23793752359108602980781449353, 2.93958790966527919227675686645, 3.23498361745316308721131452939, 3.883026539935805460662739312419, 4.772148102890967357135072331366, 5.65575747495337762262996002953, 6.03144963973624929161575905017, 6.71753350165733055122739999097, 7.594783555022568534958507762819, 8.26645900675027934379472393578, 9.28358868958116745325626448037, 9.97132945503911285509758030621, 10.60160810137954075814080980318, 11.059777938704795329899305180788, 11.75371622912998377113213485404, 12.594374937450211751863396404253, 13.39185858367575585595463733454, 13.931253308022083274471796068669, 14.43988028249312666202014998479, 15.2237923541519251620102095554, 15.66031067911261051342617748574, 16.14168840258185560241140400869, 16.8706039847982417739113130733, 17.50034064625343502084377782613

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