Properties

Degree 1
Conductor 1021
Sign $-0.794 - 0.607i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.201 − 0.979i)2-s + (0.445 + 0.895i)3-s + (−0.918 + 0.395i)4-s + (0.786 − 0.617i)5-s + (0.786 − 0.617i)6-s + (−0.542 − 0.840i)7-s + (0.572 + 0.819i)8-s + (−0.602 + 0.798i)9-s + (−0.763 − 0.645i)10-s + (0.237 − 0.971i)11-s + (−0.763 − 0.645i)12-s + (0.739 − 0.673i)13-s + (−0.713 + 0.700i)14-s + (0.903 + 0.429i)15-s + (0.687 − 0.726i)16-s + (−0.999 + 0.0369i)17-s + ⋯
L(s,χ)  = 1  + (−0.201 − 0.979i)2-s + (0.445 + 0.895i)3-s + (−0.918 + 0.395i)4-s + (0.786 − 0.617i)5-s + (0.786 − 0.617i)6-s + (−0.542 − 0.840i)7-s + (0.572 + 0.819i)8-s + (−0.602 + 0.798i)9-s + (−0.763 − 0.645i)10-s + (0.237 − 0.971i)11-s + (−0.763 − 0.645i)12-s + (0.739 − 0.673i)13-s + (−0.713 + 0.700i)14-s + (0.903 + 0.429i)15-s + (0.687 − 0.726i)16-s + (−0.999 + 0.0369i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.794 - 0.607i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.794 - 0.607i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $-0.794 - 0.607i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (256, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ -0.794 - 0.607i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3719668510 - 1.098537905i$
$L(\frac12,\chi)$  $\approx$  $0.3719668510 - 1.098537905i$
$L(\chi,1)$  $\approx$  0.8598185285 - 0.5076109848i
$L(1,\chi)$  $\approx$  0.8598185285 - 0.5076109848i

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

−22.096292716050682468839080493945, −21.28848528997958504368367494094, −20.14395749039941570732210878982, −19.20925609271883935796161352526, −18.42246793925000794293508477231, −18.28609542523563133597185956892, −17.29099754997310768933127145508, −16.583608314151166645235727894784, −15.3010275612830065886667952190, −14.92151793606251062994214986955, −14.117669534575676620642007894778, −13.27096420750340367881762338071, −12.80998551918663545141057434582, −11.71367712701022295640967483455, −10.4429543552064252630858383765, −9.440345968513240704911354263746, −8.97016027919355095611378970211, −8.1141668694327987737842289938, −6.95746799891109467800539579375, −6.47327499247098464664897765013, −6.00776316259991145825908308699, −4.73595636136450941358635923781, −3.478829971468620786627290258762, −2.280578274415811227971336015861, −1.53575427541209636456811164508, 0.4873200144803015555327339599, 1.715124898050008163686641713028, 2.896966988142301392735554811706, 3.58870130489257678466737894062, 4.45922364305554849889682460865, 5.30345939263302321837796293440, 6.36218026673053790344310452673, 7.89617506358243684759714442490, 8.78554064839880225600276539098, 9.240759030010872393213202403351, 10.02341153815866505587843622220, 10.93674906193409929861166543381, 11.202566323576278162384263492968, 12.8578435899695746881341675781, 13.39296582617588240837980172840, 13.73868101874464821842219062303, 14.89113825319418897016047433149, 15.983377934582093864513885550339, 16.78700502378272455648347284738, 17.25039710353134335533244443961, 18.2191421578572939538235539918, 19.34121863993123769253006210994, 19.91702419570948417814495924034, 20.49384777415969500968132241774, 21.186164678331772402778552412325

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