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Properties

Label 1-1001-1001.800-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.630 - 0.776i$
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  + (0.743 − 0.669i)2-s + (−0.809 + 0.587i)3-s + (0.104 − 0.994i)4-s + (0.743 + 0.669i)5-s + (−0.207 + 0.978i)6-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + 10-s + (0.5 + 0.866i)12-s + (−0.994 − 0.104i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.406 − 0.913i)18-s + (0.587 + 0.809i)19-s + (0.743 − 0.669i)20-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (−0.809 + 0.587i)3-s + (0.104 − 0.994i)4-s + (0.743 + 0.669i)5-s + (−0.207 + 0.978i)6-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + 10-s + (0.5 + 0.866i)12-s + (−0.994 − 0.104i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.406 − 0.913i)18-s + (0.587 + 0.809i)19-s + (0.743 − 0.669i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.630 - 0.776i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (800, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.630 - 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.785686873 - 0.8499550509i\)
\(L(\frac12)\) \(\approx\) \(1.785686873 - 0.8499550509i\)
\(L(1)\) \(\approx\) \(1.364141450 - 0.3636222852i\)
\(L(1)\) \(\approx\) \(1.364141450 - 0.3636222852i\)

Euler product

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

−21.91466772525708711250539096305, −21.31629248397044058772436289057, −20.4143359445215253617009493131, −19.42986295303155988676631909736, −18.23899515648279783243317777809, −17.67097810518306605799087128679, −16.908982855118318582313676578725, −16.4802508379863362653258900850, −15.55510579790661984058947972017, −14.55815177327204482323335032000, −13.57029045916297319367238527490, −13.160441320351645490150329934764, −12.398901608883008381293699995165, −11.687938396101888306410777054108, −10.74643766525321063279870100865, −9.576980209808586162998599300591, −8.58669077251022335272854013030, −7.66062453977996745604403334912, −6.85641160025737107310444506932, −5.99925486587449015617343262002, −5.32258508044918359561659201383, −4.76065878450293396456183403349, −3.47025585567229451796997650143, −2.20335278870779719360633280019, −1.13060718754053475569777792571, 0.86367658643775453701304455569, 2.10600024964476617510029812226, 3.16142272959134859367769072962, 3.93774729268638636674622671001, 5.10759276461729398973725783756, 5.613831204146320361356123364206, 6.448308221394897470551854704466, 7.284379029290328121581024656962, 9.066949079396874719330218780211, 9.82369692927717410332173362166, 10.4198608076336583128943554415, 11.085568706700707162022860527818, 11.94411021895902287259543962430, 12.60187515039177536649391869357, 13.67119738012489122015499525670, 14.38631059181351044168653996539, 15.01953727611282350901443101322, 15.996845758602147994407467592965, 16.73267120735211774553167038201, 17.831564751005488784922247219958, 18.40643595897083528495030135355, 19.135047021256842230359824417459, 20.409336646904727718541527871946, 20.960897980366726014582029472629, 21.57672982236129863789856863014

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