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Properties

Label 1-1001-1001.223-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.966 - 0.256i$
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 + (0.104 + 0.994i)3-s + (0.104 − 0.994i)4-s + (−0.951 + 0.309i)5-s + (0.743 + 0.669i)6-s + (−0.587 − 0.809i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + 12-s + (−0.406 − 0.913i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.587 + 0.809i)18-s + (−0.406 + 0.913i)19-s + (0.207 + 0.978i)20-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (0.104 + 0.994i)3-s + (0.104 − 0.994i)4-s + (−0.951 + 0.309i)5-s + (0.743 + 0.669i)6-s + (−0.587 − 0.809i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + 12-s + (−0.406 − 0.913i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.587 + 0.809i)18-s + (−0.406 + 0.913i)19-s + (0.207 + 0.978i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.966 - 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.809608058 - 0.2356663227i\)
\(L(\frac12)\) \(\approx\) \(1.809608058 - 0.2356663227i\)
\(L(1)\) \(\approx\) \(1.349263292 - 0.1469950216i\)
\(L(1)\) \(\approx\) \(1.349263292 - 0.1469950216i\)

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 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.406 + 0.913i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (0.951 + 0.309i)T \)
37 \( 1 + (0.406 + 0.913i)T \)
41 \( 1 + (0.994 - 0.104i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.994 + 0.104i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.743 - 0.669i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.207 - 0.978i)T \)
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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

−21.78396730593945830897639953903, −21.00666909357062979476666267557, −19.99384520789567630948881164603, −19.44020270635531901816509490721, −18.59665499485901835462383807377, −17.485815678846325656952642915010, −17.068856528713065515756346742113, −16.02933100936920316277741625125, −15.304388442387696261388232189688, −14.5750222934621218523493214701, −13.71721476926941465340701417061, −12.927785148662902470990376220402, −12.3372660171178673997766792745, −11.66464800229395141762124105879, −10.837075192693881971670803236748, −9.07778303219214731408892761025, −8.40687544486332435603160758247, −7.62818798307403508902125865504, −7.06297700434062198179827068382, −6.1229559930734270738356849085, −5.24345027753773052432896561792, −4.21885930780225060575610004057, −3.31914295919535204882939068886, −2.382677584604085446675196224126, −0.87875793896660180710239351819, 0.859763444408236859960454723936, 2.63495045963210123777736668358, 3.12954644327963742862943356129, 4.21306087067114475075527588327, 4.60502088057973685100828630331, 5.70844813305208024335828132477, 6.64232420024929128515046316232, 7.898281350724888743282535610452, 8.79654626603568193639961237216, 9.90538403331105234959094726499, 10.42340025088791748034984660716, 11.2991552243516302555201957298, 11.89678677548523471677353881270, 12.68412803240489896361797697011, 13.936145482501599618449757587216, 14.529483629295491628601153093160, 15.13826146709757502496061573027, 15.98955048588375359504277372012, 16.488995045326104758451370760399, 17.8287202678972208182264655352, 18.98737912951813704127993133204, 19.32102782122125747129085039029, 20.35996829222790508792263685056, 20.83390578965411834784749045030, 21.519964965839667181347306246227

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