Properties

Degree 1
Conductor 1021
Sign $-0.761 - 0.647i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $-0.761 - 0.647i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (71, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ -0.761 - 0.647i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.141230703 - 3.103460695i$
$L(\frac12,\chi)$  $\approx$  $1.141230703 - 3.103460695i$
$L(\chi,1)$  $\approx$  1.621485826 - 1.471662716i
$L(1,\chi)$  $\approx$  1.621485826 - 1.471662716i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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