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Properties

Label 1-2001-2001.896-r1-0-0
Degree $1$
Conductor $2001$
Sign $-0.950 + 0.310i$
Analytic cond. $215.037$
Root an. cond. $215.037$
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  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.900 − 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.974 − 0.222i)8-s + (0.781 + 0.623i)10-s + (0.974 − 0.222i)11-s + (0.222 + 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.222 − 0.974i)16-s + i·17-s + (−0.781 − 0.623i)19-s + (−0.222 + 0.974i)20-s + (0.623 + 0.781i)22-s + (0.623 − 0.781i)25-s + (−0.781 + 0.623i)26-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.900 − 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.974 − 0.222i)8-s + (0.781 + 0.623i)10-s + (0.974 − 0.222i)11-s + (0.222 + 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.222 − 0.974i)16-s + i·17-s + (−0.781 − 0.623i)19-s + (−0.222 + 0.974i)20-s + (0.623 + 0.781i)22-s + (0.623 − 0.781i)25-s + (−0.781 + 0.623i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.950 + 0.310i$
Analytic conductor: \(215.037\)
Root analytic conductor: \(215.037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (896, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (1:\ ),\ -0.950 + 0.310i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2674748116 + 1.678276083i\)
\(L(\frac12)\) \(\approx\) \(0.2674748116 + 1.678276083i\)
\(L(1)\) \(\approx\) \(1.111272477 + 0.5692967231i\)
\(L(1)\) \(\approx\) \(1.111272477 + 0.5692967231i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.433 + 0.900i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (0.974 - 0.222i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.781 - 0.623i)T \)
31 \( 1 + (-0.433 - 0.900i)T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (0.781 - 0.623i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.433 + 0.900i)T \)
79 \( 1 + (0.974 + 0.222i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.433 - 0.900i)T \)
97 \( 1 + (0.781 + 0.623i)T \)
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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

−19.33361299256508422336863281506, −18.90758485215760603438985676195, −18.07167647731805079369862791404, −17.591421063895537705779919239165, −16.61471135943109106433358429812, −15.48533313242800815299697254202, −14.95263667392632159943259679749, −14.05801972907930474889991811819, −13.64232126318316703792654534582, −12.49397228871783134790382246633, −12.38207630023345077392453526792, −11.28618449360521298607436772310, −10.47073616330325560637931620558, −9.91560739518110494890696720009, −9.125937542561601734412718093996, −8.62718820208596749371999229755, −7.07825265718242701442122661824, −6.28283659701115967882865473128, −5.64130936084346774290563995602, −4.94994731226856138165436692155, −3.68524571987382626719127449858, −3.08785295403405277415945097760, −2.22976934714125090508218088050, −1.49490994617440348493308177398, −0.26199292878737692606895953434, 0.98862971878928732062674306769, 2.053423965261382287909186022838, 3.35116273065276394026457907672, 4.14052792002136606880298295490, 4.711856862303771023262353002446, 5.94763294423074368911504827506, 6.37714185753779055989089009300, 6.94839079968824771155228633847, 8.03560757966936151175929890239, 8.99266545371568326671051963199, 9.31606726266816013996223771098, 10.28477103588325355929868742787, 11.28760307418770814851275345407, 12.30628408529296776203277883748, 13.045255801088923531107163906004, 13.51239533635353835414385760053, 14.25194313583893345689173385611, 14.80226415887467907585785385728, 15.8808073880903394160942579608, 16.56991426217289722675903237012, 17.13950329193743447429587251028, 17.39420315079394185098280773309, 18.56541081289606311333587571914, 19.31557730045427968736880398784, 20.15970678342924868545966969032

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