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Properties

Label 1-1001-1001.115-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.955 - 0.294i$
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  + (−0.406 + 0.913i)2-s + (−0.309 − 0.951i)3-s + (−0.669 − 0.743i)4-s + (−0.406 − 0.913i)5-s + (0.994 + 0.104i)6-s + (0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + 10-s + (−0.5 + 0.866i)12-s + (−0.743 + 0.669i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.207 − 0.978i)18-s + (0.951 − 0.309i)19-s + (−0.406 + 0.913i)20-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.309 − 0.951i)3-s + (−0.669 − 0.743i)4-s + (−0.406 − 0.913i)5-s + (0.994 + 0.104i)6-s + (0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + 10-s + (−0.5 + 0.866i)12-s + (−0.743 + 0.669i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.207 − 0.978i)18-s + (0.951 − 0.309i)19-s + (−0.406 + 0.913i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.955 - 0.294i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.955 - 0.294i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9147083834 - 0.1379052741i\)
\(L(\frac12)\) \(\approx\) \(0.9147083834 - 0.1379052741i\)
\(L(1)\) \(\approx\) \(0.7474453537 - 0.03492486087i\)
\(L(1)\) \(\approx\) \(0.7474453537 - 0.03492486087i\)

Euler product

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

−21.578592694518544518388461409671, −20.97284200052903675288090892810, −20.211863277824213575294464053236, −19.33463861853693284365020269892, −18.70096666611949791656856905314, −17.82382365539886816927480202146, −17.14776372679890914500758815858, −16.21982593550109731960433911791, −15.53569825721755897998897391619, −14.393629671845003056059998829691, −14.01851480507885741036002886521, −12.51430937412431546429278194640, −11.91534836029461735549989886198, −11.11989983236814730664047651641, −10.41246439632440692080077027586, −9.952041492689031309476285360024, −8.95178377445472706105787647345, −8.07836193354475878518253348278, −7.13080845075790008638307865634, −5.91112563083142117971431789795, −4.8595028615035885232121568939, −3.87728013084772188386109977784, −3.27239092748517672996291013930, −2.400990082127762086978171266253, −0.77886427262294894010948503092, 0.82306427903201061983778990860, 1.42024155301821132069313570791, 3.07134898481834946092852133891, 4.54468692709374142574650300609, 5.31060499964074428202962137442, 5.98363005592850225572755106071, 7.13715645733776673924704280914, 7.64569139969806213574317703115, 8.41931659895687813068274066198, 9.2261196547860485775125634306, 10.13790100673616445001421689379, 11.45307713396653242780588269937, 12.01375854515487436571625568945, 13.15016259384687146973898143812, 13.54353653772246551078141899446, 14.542226109198741606840614693417, 15.48326095935649615514001765223, 16.39269687212103056720129073899, 16.80284269532141184954294821085, 17.68455226499695559624583393447, 18.327830078952093695290210822939, 19.163431987168117307370399046752, 19.7706877793022237943547366654, 20.56985763324653876061621962896, 21.832777350998611667244643197293

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