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Properties

Label 1-1001-1001.236-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.578 + 0.815i$
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  + (0.587 + 0.809i)2-s + (−0.669 + 0.743i)3-s + (−0.309 + 0.951i)4-s + (−0.994 − 0.104i)5-s + (−0.994 − 0.104i)6-s + (−0.951 + 0.309i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.743 − 0.669i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.743 − 0.669i)18-s + (0.207 − 0.978i)19-s + (0.406 − 0.913i)20-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.669 + 0.743i)3-s + (−0.309 + 0.951i)4-s + (−0.994 − 0.104i)5-s + (−0.994 − 0.104i)6-s + (−0.951 + 0.309i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.743 − 0.669i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.743 − 0.669i)18-s + (0.207 − 0.978i)19-s + (0.406 − 0.913i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.578 + 0.815i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.578 + 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7998963149 + 0.4131953842i\)
\(L(\frac12)\) \(\approx\) \(0.7998963149 + 0.4131953842i\)
\(L(1)\) \(\approx\) \(0.7064220044 + 0.4681267848i\)
\(L(1)\) \(\approx\) \(0.7064220044 + 0.4681267848i\)

Euler product

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

−21.72178412293088527957698327980, −20.713479572587418998694848128658, −19.69015263922042482148410087771, −19.47211811409248375741362291724, −18.4839426457503258288531355935, −17.98530090104989487770067699880, −16.87536769829610107509939981363, −15.89733744894456285848174187159, −15.18843906424845990015225936288, −14.13860425815187347111916821251, −13.4820070563224335976280303643, −12.34899971570483170790439905444, −12.21522905502687430768860041474, −11.23039363672513129292889230602, −10.70823033802151293423245895990, −9.726090106667507926077364423311, −8.37163287672279605218500201568, −7.70949051975048701532969075763, −6.45453698519989845776351489006, −5.96994865239766126867057083676, −4.69583199749638279040361197376, −4.1213847499606910250407878895, −2.94421461504827940169034372122, −1.93052118962667630676393008617, −0.8349501915085281529439102275, 0.48697104196929593976920888, 2.76943958706336336674063430024, 3.67922898361328816939964775349, 4.59301290586596054923292589787, 4.939732549967233629566295945296, 6.1867080539186315679030056041, 6.847588475717948899022975060274, 7.815531841926399515544712445524, 8.72687006324037237696662145560, 9.498188683873703459432865736530, 10.759720608793141927174381780135, 11.59298730784438524015445645247, 12.08209087008769319393525850706, 13.03544002629764120335543602809, 14.030370697408271377230797151582, 14.95017177475038554516438751299, 15.63994689325857362010206000842, 16.03924095484361241741178777300, 16.76707506046233969358465222799, 17.73575457964815870359211052342, 18.2312686164202288461852294501, 19.645399955821782111680488401981, 20.35169618176006712572764090170, 21.25153316609768052806967025461, 22.03160490954695846732491924092

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