↑

Properties

Label 1-1001-1001.878-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

Related objects

Downloads

Learn more

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: $\chi_{1001} (878, \cdot )$
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 \( 1 + (-0.406 - 0.913i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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