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Properties

Label 1-1001-1001.160-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.0698 + 0.997i$
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  + (0.913 + 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (−0.913 + 0.406i)6-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s − 12-s + (0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (0.309 − 0.951i)18-s + (0.978 + 0.207i)19-s + (−0.104 − 0.994i)20-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (−0.913 + 0.406i)6-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s − 12-s + (0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (0.309 − 0.951i)18-s + (0.978 + 0.207i)19-s + (−0.104 − 0.994i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0698 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0698 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.0698 + 0.997i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.0698 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.338931976 + 1.248452603i\)
\(L(\frac12)\) \(\approx\) \(1.338931976 + 1.248452603i\)
\(L(1)\) \(\approx\) \(1.215172578 + 0.5954218333i\)
\(L(1)\) \(\approx\) \(1.215172578 + 0.5954218333i\)

Euler product

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

−21.91696087893780403639988445378, −20.68538809990827989256413192317, −19.77488646771321519737456277504, −19.25801207875040806233530049868, −18.5110326713521250636063812083, −17.772461135808699884838355938165, −16.50935434967655499478046320654, −15.94474190921165324495817060048, −14.99965679666017188492196965365, −14.20174213028475703886345740415, −13.47151838505777146744265700969, −12.54440130117952308873882125208, −11.88984955392052652723325715767, −11.36812567710682438443053802653, −10.579293930849774464498164167216, −9.7202900463396217446125831591, −8.03599958113050685561099282491, −7.398802006477003656722644935665, −6.59246000216496774559266519565, −5.74571296767639032151589591232, −4.94809687905193109438554844174, −3.82788908518135291967535116455, −3.02088424054539881697604685345, −1.91806458606253108040634956366, −0.78932904509245554336693253206, 1.04272835155901415622786375741, 2.866279404127683887460225600, 3.72769059723160000717977735385, 4.480101140019207034927945752095, 5.20440990190238729719256392440, 5.931209014151221128336624700053, 7.00448283508414131933436325171, 7.86156289631381457465109359781, 8.7680462011272626369523620889, 9.8618369804776761620737818992, 10.869038542526156556292089180654, 11.741061690986409403423043277574, 12.16106081203761994135564046157, 12.93075552093842235701509585164, 14.17684715832624735603043550909, 14.74610311364756324485054932387, 15.83649282504828396838815372286, 16.083222146854498709396007504882, 16.75366140722677546267805071300, 17.61494809084874551879301920285, 18.63533439694446041218745366285, 19.94516067807672893901527651436, 20.49450735408493629794816737729, 21.12594539481542667147071004337, 22.06570225252434112048976266202

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