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Properties

Label 1-1001-1001.821-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.527 - 0.849i$
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  + (−0.951 + 0.309i)2-s + (−0.104 + 0.994i)3-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (−0.207 − 0.978i)6-s + (−0.587 + 0.809i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.994 + 0.104i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.994 − 0.104i)18-s + (0.406 + 0.913i)19-s + (0.743 + 0.669i)20-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (−0.104 + 0.994i)3-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (−0.207 − 0.978i)6-s + (−0.587 + 0.809i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.994 + 0.104i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.994 − 0.104i)18-s + (0.406 + 0.913i)19-s + (0.743 + 0.669i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.527 - 0.849i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.527 - 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1603040357 + 0.2881900437i\)
\(L(\frac12)\) \(\approx\) \(-0.1603040357 + 0.2881900437i\)
\(L(1)\) \(\approx\) \(0.4408293717 + 0.3590235850i\)
\(L(1)\) \(\approx\) \(0.4408293717 + 0.3590235850i\)

Euler product

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

−20.87267480790432646164402129565, −20.112404866707062885649515215, −19.53603205583037805526500647451, −18.85115599098285180068237881527, −17.89276582816561283097286587733, −17.40353276984017838765577960310, −16.74168440233744166631469893636, −15.94782060118130028476118959229, −14.93156643005834453538503537683, −13.60389912808777314948253639322, −13.081285378639825503867406102398, −12.14524499410934288210678699751, −11.73894364300026582887866645640, −10.68494642022721475769726179109, −9.67847516358464364748798024824, −8.90360244631603647406480002580, −8.079527758324609076138270471902, −7.59187679883118629404848884015, −6.37882234530789909628496836406, −5.79010089298138555421453442521, −4.430773958468700875807266510706, −3.11556276939368727159678689792, −2.00860819125940812980010260886, −1.34496504968150452910783182224, −0.19250212018309403057063484436, 1.653296643249490080963495473362, 2.8544945960487732962624402575, 3.55854428500688014786578673534, 5.03603794785186598048321963713, 5.81541100644330625861362763605, 6.67554972334982956356680772864, 7.582461346506944897883527829978, 8.48957387713874538835596845909, 9.502349302415854055958095018107, 10.0062750520211526833678800299, 10.71439450953585249309199658996, 11.41927701123409492684443285016, 12.240113093517380985297598778448, 14.16517231729546890014504420108, 14.26097945898759609000718680400, 15.269392782102752910722531783886, 16.074435459359553162765408972891, 16.486451703903280768245640066953, 17.62638463789196283208915962309, 18.095064959801164538937229085392, 18.894607740859734204528099105014, 19.8539386199076132512793479895, 20.49525024621364836687675709533, 21.34718643254347864789444578535, 22.09050460269079125277719813286

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