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Properties

Label 1-2001-2001.227-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.983 + 0.181i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
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  + (0.0611 + 0.998i)2-s + (−0.992 + 0.122i)4-s + (0.996 + 0.0815i)5-s + (−0.0203 − 0.999i)7-s + (−0.182 − 0.983i)8-s + (−0.0203 + 0.999i)10-s + (−0.377 − 0.925i)11-s + (−0.452 + 0.891i)13-s + (0.996 − 0.0815i)14-s + (0.970 − 0.242i)16-s + (0.841 + 0.540i)17-s + (0.992 − 0.122i)19-s + (−0.999 + 0.0407i)20-s + (0.900 − 0.433i)22-s + (0.986 + 0.162i)25-s + (−0.917 − 0.396i)26-s + ⋯
L(s)  = 1  + (0.0611 + 0.998i)2-s + (−0.992 + 0.122i)4-s + (0.996 + 0.0815i)5-s + (−0.0203 − 0.999i)7-s + (−0.182 − 0.983i)8-s + (−0.0203 + 0.999i)10-s + (−0.377 − 0.925i)11-s + (−0.452 + 0.891i)13-s + (0.996 − 0.0815i)14-s + (0.970 − 0.242i)16-s + (0.841 + 0.540i)17-s + (0.992 − 0.122i)19-s + (−0.999 + 0.0407i)20-s + (0.900 − 0.433i)22-s + (0.986 + 0.162i)25-s + (−0.917 − 0.396i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.983 + 0.181i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.983 + 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.694598045 + 0.1551982312i\)
\(L(\frac12)\) \(\approx\) \(1.694598045 + 0.1551982312i\)
\(L(1)\) \(\approx\) \(1.122227339 + 0.3319962028i\)
\(L(1)\) \(\approx\) \(1.122227339 + 0.3319962028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.0611 + 0.998i)T \)
5 \( 1 + (0.996 + 0.0815i)T \)
7 \( 1 + (-0.0203 - 0.999i)T \)
11 \( 1 + (-0.377 - 0.925i)T \)
13 \( 1 + (-0.452 + 0.891i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (0.992 - 0.122i)T \)
31 \( 1 + (-0.591 - 0.806i)T \)
37 \( 1 + (-0.685 - 0.728i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (0.591 - 0.806i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (0.794 + 0.607i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (-0.262 + 0.965i)T \)
67 \( 1 + (0.377 - 0.925i)T \)
71 \( 1 + (0.818 + 0.574i)T \)
73 \( 1 + (-0.714 - 0.699i)T \)
79 \( 1 + (-0.970 - 0.242i)T \)
83 \( 1 + (0.557 - 0.830i)T \)
89 \( 1 + (0.882 - 0.470i)T \)
97 \( 1 + (0.301 - 0.953i)T \)
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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

−20.1432140557297255633820986843, −19.18527235710106875350249571079, −18.406954638003731515414142492427, −17.89725473749079947951292906043, −17.46261894273023253858260817907, −16.36469465551204254688360573040, −15.39826331643413675661657639064, −14.58179704836980718573938919918, −14.01416270443620318188388293262, −13.080597097743124434023129138112, −12.4599527671527096600731334145, −12.04951601036312301382701844289, −10.99477786172315499329841644701, −10.06170364881406353455814790187, −9.74275802534653646118839641334, −9.014825483994568124384841350535, −8.11332009399082630606904664288, −7.17620945772100178818381060228, −5.84265672048660788338163412748, −5.28214188032326003432864804607, −4.81104598095226005404457630175, −3.32501515423939759783372890769, −2.72135669969187214795898967318, −1.97589763117491587575751713920, −1.074485373922078967542744635847, 0.65639570940437307251270282452, 1.71936594433364939355029934061, 3.10810109591543897356389936694, 3.88133923623306247917024106153, 4.89182669223430947445872665526, 5.62430569913958900692932029750, 6.27444039139106589132647022349, 7.16496112840322247081482127082, 7.70499671358528306688020604820, 8.7209467377452343364306842290, 9.47665703766436709277155035013, 10.105153370012112958396039756168, 10.85986944571078676345847502729, 11.985726658452091814128684340043, 13.05770395124320035215689740602, 13.52527043033891145381316753358, 14.23048031620664080961304212142, 14.564075637460919255982473868468, 15.753496377323900908239389840045, 16.6178117329527313885351560331, 16.80673491300700247483490489030, 17.60349486623953581895385610783, 18.40835813410386810130900068306, 18.96468483783581487941655364587, 19.90207932137237177440883761090

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