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Properties

Label 1-1021-1021.71-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.761 - 0.647i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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  + (0.786 − 0.617i)2-s + (0.932 − 0.361i)3-s + (0.237 − 0.971i)4-s + (0.510 − 0.859i)5-s + (0.510 − 0.859i)6-s + (0.0184 + 0.999i)7-s + (−0.412 − 0.911i)8-s + (0.739 − 0.673i)9-s + (−0.128 − 0.991i)10-s + (−0.478 − 0.878i)11-s + (−0.128 − 0.991i)12-s + (−0.273 − 0.961i)13-s + (0.631 + 0.775i)14-s + (0.165 − 0.986i)15-s + (−0.886 − 0.462i)16-s + (−0.918 + 0.395i)17-s + ⋯
L(s)  = 1  + (0.786 − 0.617i)2-s + (0.932 − 0.361i)3-s + (0.237 − 0.971i)4-s + (0.510 − 0.859i)5-s + (0.510 − 0.859i)6-s + (0.0184 + 0.999i)7-s + (−0.412 − 0.911i)8-s + (0.739 − 0.673i)9-s + (−0.128 − 0.991i)10-s + (−0.478 − 0.878i)11-s + (−0.128 − 0.991i)12-s + (−0.273 − 0.961i)13-s + (0.631 + 0.775i)14-s + (0.165 − 0.986i)15-s + (−0.886 − 0.462i)16-s + (−0.918 + 0.395i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.761 - 0.647i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.761 - 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141230703 - 3.103460695i\)
\(L(\frac12)\) \(\approx\) \(1.141230703 - 3.103460695i\)
\(L(1)\) \(\approx\) \(1.621485826 - 1.471662716i\)
\(L(1)\) \(\approx\) \(1.621485826 - 1.471662716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.786 - 0.617i)T \)
3 \( 1 + (0.932 - 0.361i)T \)
5 \( 1 + (0.510 - 0.859i)T \)
7 \( 1 + (0.0184 + 0.999i)T \)
11 \( 1 + (-0.478 - 0.878i)T \)
13 \( 1 + (-0.273 - 0.961i)T \)
17 \( 1 + (-0.918 + 0.395i)T \)
19 \( 1 + (0.739 + 0.673i)T \)
23 \( 1 + (-0.886 + 0.462i)T \)
29 \( 1 + (0.0184 - 0.999i)T \)
31 \( 1 + (-0.201 + 0.979i)T \)
37 \( 1 + (-0.0554 - 0.998i)T \)
41 \( 1 + (0.739 + 0.673i)T \)
43 \( 1 + (0.956 + 0.291i)T \)
47 \( 1 + (0.631 + 0.775i)T \)
53 \( 1 + (0.989 + 0.147i)T \)
59 \( 1 + (-0.542 + 0.840i)T \)
61 \( 1 + (0.631 + 0.775i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.412 - 0.911i)T \)
73 \( 1 + (0.631 - 0.775i)T \)
79 \( 1 + (0.687 + 0.726i)T \)
83 \( 1 + (0.932 - 0.361i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (-0.602 - 0.798i)T \)
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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

−22.1944698389011096709353154591, −21.18493046656072589208873002850, −20.393489806939577602899881069735, −19.98899112913139925088035901903, −18.71269953209841782752376493567, −17.88407285361178485736470950314, −17.111821390523305658587449726702, −16.12154186336462623858246451709, −15.487170545272096335449570280791, −14.63741863617183864775877467986, −14.04905569629082615672004154052, −13.57611521826796728025166574397, −12.803478028775145036556483807252, −11.5332261175834164627565600984, −10.65732321867870905864074710988, −9.79123003381242380218916119544, −8.99431667782466469220120780828, −7.77281609152903721097197618176, −7.11700737847562783638285971324, −6.6895721380626016661271071243, −5.23704337590586121239746675408, −4.379278307456550969581343752626, −3.73780281841863994096753207891, −2.59124862735902555637335844385, −2.05717510619283548434400031542, 0.90937949587809336535215313534, 2.01566407552512311839238783439, 2.633783072692371991669980125529, 3.57660157435588550797476879409, 4.626132999682207269329072173123, 5.76364082894831465206284379769, 5.99990737308028564930333107141, 7.61704442791080733193699570583, 8.47439614782558353552458144155, 9.24263679225913627106258851356, 9.94509227474303193329963767316, 10.980303912309812214984156018457, 12.24819933352541425978702151783, 12.49611138778020063896942454233, 13.44603270954888228545652731756, 13.86931350857243163463039282790, 14.84600643125483859096062418289, 15.64468281563009398363023420501, 16.16804721457488520310789517493, 17.87895130498431072158235130482, 18.18156181841843879763187854414, 19.41174878422358668442555672169, 19.70760458038798233339321058344, 20.690500160832684317331449789018, 21.20704010810764061306242509247

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