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Properties

Label 1-6001-6001.70-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.165 - 0.986i$
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  − i·2-s − 3-s − 4-s − 5-s + i·6-s − 7-s + i·8-s + 9-s + i·10-s + (−0.707 − 0.707i)11-s + 12-s + (0.707 + 0.707i)13-s + i·14-s + 15-s + 16-s + ⋯
L(s)  = 1  − i·2-s − 3-s − 4-s − 5-s + i·6-s − 7-s + i·8-s + 9-s + i·10-s + (−0.707 − 0.707i)11-s + 12-s + (0.707 + 0.707i)13-s + i·14-s + 15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.165 - 0.986i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.165 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5342279951 - 0.4518862300i\)
\(L(\frac12)\) \(\approx\) \(0.5342279951 - 0.4518862300i\)
\(L(1)\) \(\approx\) \(0.4950945387 - 0.2792878655i\)
\(L(1)\) \(\approx\) \(0.4950945387 - 0.2792878655i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 - T \)
47 \( 1 - iT \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + iT \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−17.894120200868540813278396485591, −17.02644459436748755233909224637, −16.45033507373475030728616205589, −15.814819315268187474718314223507, −15.48370745935206771774891556865, −15.10738502789694892100406603830, −13.80239800390380841584388818490, −13.238567609192142231262086049069, −12.62899857086115032871339406888, −12.11056097245395439352647656764, −11.32435106283112777378542843227, −10.33561340608220230939307224185, −10.04554628134941027135488942561, −9.14550101582462119929991439312, −8.27072038450203115289531089864, −7.62787596522461945826667329686, −6.96593757022434232961030088081, −6.556356083072035069497911259386, −5.63642392069246334485828311654, −5.06788790460684528205028252794, −4.476565063414312631471140439690, −3.50003571591903932892649335713, −3.06529962860446168973375140227, −1.25646769597058159899812175361, −0.52006889505476021710587504475, 0.55445233358004475909278110927, 1.01266244239420206631642400685, 2.32444542752163101786535700919, 3.24121673552489618684652601550, 3.675849506758631659957798432016, 4.54754966764349027435159376785, 5.10230942809372555465830845247, 6.02135162771997259909675706096, 6.65060044128542932724108258167, 7.51945321804711837390959427138, 8.34904572331916103256383760725, 9.00354882943829032785265881950, 9.85843287928733952073564158188, 10.48062656791771764794492948783, 10.99486897049357204647920413912, 11.82720653369963249126122897553, 11.92854999141767392475878185545, 12.85934068028377906579425837056, 13.37468869718669094503858146796, 13.94689890022963505584161984145, 15.16840329858331670922029903258, 15.71665058339925428319184320296, 16.51134117777591802837215194311, 16.655348594372319109491736225, 17.80583232021271154399344190642

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