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Properties

Label 1-1001-1001.114-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.722 - 0.691i$
Analytic cond. $4.64862$
Root an. cond. $4.64862$
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  + (−0.309 − 0.951i)2-s + (−0.104 + 0.994i)3-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)5-s + (0.978 − 0.207i)6-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.104 + 0.994i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.104 + 0.994i)18-s + (−0.913 + 0.406i)19-s + (−0.669 + 0.743i)20-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.104 + 0.994i)3-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)5-s + (0.978 − 0.207i)6-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.104 + 0.994i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.104 + 0.994i)18-s + (−0.913 + 0.406i)19-s + (−0.669 + 0.743i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.722 - 0.691i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.722 - 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.186674465 - 0.4763360996i\)
\(L(\frac12)\) \(\approx\) \(1.186674465 - 0.4763360996i\)
\(L(1)\) \(\approx\) \(0.9493296887 - 0.2028062047i\)
\(L(1)\) \(\approx\) \(0.9493296887 - 0.2028062047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-0.104 + 0.994i)T \)
5 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.104 - 0.994i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−21.98357559664823256667715164565, −21.08032375305686144246019237473, −19.865277862210515846687583844155, −19.05515618232331693894133675663, −18.53336426212353034714742739393, −17.68792956073299769331854657141, −17.13123754499146508936070706042, −16.65175755605766046045037649514, −15.28489329119698564346688632695, −14.606854474177998677694464646638, −13.90976197073196293041073564867, −13.06458756589819537620401636019, −12.6686148863644696716888104404, −11.17625570493564811883740109268, −10.41670929186946280281722360489, −9.42458449981913991184633300385, −8.587833133224302604415690254196, −7.88865500521484906133991527781, −6.738355725764622861712945166029, −6.44328531801440950585603143536, −5.54550001765128355930842613918, −4.697724959122940876795664973791, −3.11490209525117684015352015812, −1.90382108408093738273404954374, −1.03888810740160722836501861009, 0.76031589196202114462124893449, 2.14174418284992716213790103762, 2.9050725102311568133815809857, 3.951711251419522456059894949043, 4.87119364742894187102850682888, 5.51632532308456219028500620657, 6.72271666014564493621372794731, 8.21232577794851847788798907182, 8.87162219852249973058885482012, 9.7738143930717688952930619795, 10.078390353581696921112755527132, 11.038879056678252896896591159348, 11.74753494680688393820648961886, 12.73928101607340674133125051405, 13.582889699485037979160582210105, 14.27991916149604794610694702250, 15.19429974049082042671373888813, 16.38105353371913695892010590504, 16.988571392070647234703555287708, 17.548770399853582703805443605939, 18.47951426813982604645124511949, 19.2865677181003127063169455081, 20.33048954560684388561498832469, 20.85413731123497725705419880934, 21.36426451641807860342267586861

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