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Properties

Label 1-2001-2001.1145-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.182 + 0.983i$
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  + (0.940 + 0.339i)2-s + (0.768 + 0.639i)4-s + (−0.0611 + 0.998i)5-s + (0.917 − 0.396i)7-s + (0.505 + 0.862i)8-s + (−0.396 + 0.917i)10-s + (0.994 + 0.101i)11-s + (0.999 − 0.0407i)13-s + (0.998 − 0.0611i)14-s + (0.182 + 0.983i)16-s + (0.909 + 0.415i)17-s + (−0.639 + 0.768i)19-s + (−0.685 + 0.728i)20-s + (0.900 + 0.433i)22-s + (−0.992 − 0.122i)25-s + (0.953 + 0.301i)26-s + ⋯
L(s)  = 1  + (0.940 + 0.339i)2-s + (0.768 + 0.639i)4-s + (−0.0611 + 0.998i)5-s + (0.917 − 0.396i)7-s + (0.505 + 0.862i)8-s + (−0.396 + 0.917i)10-s + (0.994 + 0.101i)11-s + (0.999 − 0.0407i)13-s + (0.998 − 0.0611i)14-s + (0.182 + 0.983i)16-s + (0.909 + 0.415i)17-s + (−0.639 + 0.768i)19-s + (−0.685 + 0.728i)20-s + (0.900 + 0.433i)22-s + (−0.992 − 0.122i)25-s + (0.953 + 0.301i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.182 + 0.983i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.182 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.048257946 + 2.535099440i\)
\(L(\frac12)\) \(\approx\) \(3.048257946 + 2.535099440i\)
\(L(1)\) \(\approx\) \(2.072916649 + 0.9476150084i\)
\(L(1)\) \(\approx\) \(2.072916649 + 0.9476150084i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.940 + 0.339i)T \)
5 \( 1 + (-0.0611 + 0.998i)T \)
7 \( 1 + (0.917 - 0.396i)T \)
11 \( 1 + (0.994 + 0.101i)T \)
13 \( 1 + (0.999 - 0.0407i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (-0.639 + 0.768i)T \)
31 \( 1 + (-0.0815 - 0.996i)T \)
37 \( 1 + (0.574 - 0.818i)T \)
41 \( 1 + (-0.281 + 0.959i)T \)
43 \( 1 + (0.0815 - 0.996i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (-0.882 - 0.470i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (-0.830 - 0.557i)T \)
67 \( 1 + (-0.101 - 0.994i)T \)
71 \( 1 + (0.947 - 0.320i)T \)
73 \( 1 + (-0.202 - 0.979i)T \)
79 \( 1 + (-0.983 - 0.182i)T \)
83 \( 1 + (-0.742 + 0.670i)T \)
89 \( 1 + (-0.359 - 0.933i)T \)
97 \( 1 + (0.162 + 0.986i)T \)
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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

−20.01469373660963222936767671988, −19.28040801368508234875247749484, −18.49504933805180519828656161115, −17.49530297485249521405277952154, −16.772992880399888988559918970643, −15.97207626618872754445573541269, −15.393671829769985011230276873205, −14.39896148544963743478158039595, −14.042929653856008438083799924215, −13.04585455976826891591464475728, −12.505831319505551318687701218288, −11.49755767413034267966361143331, −11.45231217328225101629686814772, −10.27667159374212141896253240806, −9.26900390035012050977501958434, −8.62037590579561418496953748236, −7.76973721295283823925005614116, −6.66547656678527612627879755464, −5.88998874837946989595717700433, −5.10730474807265134987204458025, −4.48297972393206793334163463521, −3.73303867939838529330544195632, −2.71115825079497328580761728605, −1.47886476243416440114037823951, −1.1650838345384017332984002534, 1.4379967338681221560140775317, 2.15345654202770265022167733710, 3.480058937361187874446898267903, 3.800369087093202587500575845430, 4.68608890939169166947122757020, 5.89246381304200432547175122340, 6.253136024285353179663825706954, 7.21019355192794928226797563578, 7.877874508554991081453772133078, 8.553926970779418100729409741704, 9.91111714733780945303785557570, 10.76694104328310356686004548842, 11.322221595476949564597118624486, 11.930128565304809088459894123348, 12.86192207167790033858090979332, 13.78336785976268410252367634888, 14.32785932493759867252354622949, 14.82690120183399386322885962119, 15.40461371569560224055660563161, 16.56158575901655954141567799793, 16.96870769704740664015319600360, 17.87999416622203766977247637712, 18.59323189343109422013282236983, 19.49772654260597921994079298096, 20.27798657115585545052945793200

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