Properties

Degree 1
Conductor 1021
Sign $0.819 + 0.573i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.687 − 0.726i)2-s + (−0.273 − 0.961i)3-s + (−0.0554 − 0.998i)4-s + (−0.886 − 0.462i)5-s + (−0.886 − 0.462i)6-s + (−0.659 − 0.751i)7-s + (−0.763 − 0.645i)8-s + (−0.850 + 0.526i)9-s + (−0.945 + 0.326i)10-s + (0.572 + 0.819i)11-s + (−0.945 + 0.326i)12-s + (−0.982 − 0.183i)13-s + (−0.999 − 0.0369i)14-s + (−0.201 + 0.979i)15-s + (−0.993 + 0.110i)16-s + (0.989 − 0.147i)17-s + ⋯
L(s,χ)  = 1  + (0.687 − 0.726i)2-s + (−0.273 − 0.961i)3-s + (−0.0554 − 0.998i)4-s + (−0.886 − 0.462i)5-s + (−0.886 − 0.462i)6-s + (−0.659 − 0.751i)7-s + (−0.763 − 0.645i)8-s + (−0.850 + 0.526i)9-s + (−0.945 + 0.326i)10-s + (0.572 + 0.819i)11-s + (−0.945 + 0.326i)12-s + (−0.982 − 0.183i)13-s + (−0.999 − 0.0369i)14-s + (−0.201 + 0.979i)15-s + (−0.993 + 0.110i)16-s + (0.989 − 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.819 + 0.573i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.819 + 0.573i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $0.819 + 0.573i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (108, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ 0.819 + 0.573i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1757121240 - 0.05535409646i$
$L(\frac12,\chi)$  $\approx$  $-0.1757121240 - 0.05535409646i$
$L(\chi,1)$  $\approx$  0.4646255832 - 0.6642065456i
$L(1,\chi)$  $\approx$  0.4646255832 - 0.6642065456i

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

−22.379916448233920055997933257875, −21.71074879141810705796698283756, −21.216117793055535229954113257807, −20.01153549332713759336125771683, −19.22627459006599823743771282387, −18.408673464300987559114491435397, −17.1359238893514331431132214104, −16.573234247029267556228013696229, −16.0078249047163184491173009071, −15.11095774127634624947569832137, −14.77574149224528993678433999647, −13.959611719640820328480805826475, −12.660863114574901742533150294518, −11.84130064930895100974712292149, −11.56357999816036139368694443386, −10.23063655731708780701754073513, −9.407101213630924160335287812346, −8.38700249911638183076240474617, −7.75893334945341281551234973271, −6.31170996343730556736344629673, −6.1228756103944808849632490192, −4.93062467612829912559787537262, −4.06118813400835462049347350239, −3.37981566223778956304164145992, −2.623533541358288857697058434639, 0.07035921807492099665082955552, 1.09691830029876224471335161887, 2.181270087280726201827282429889, 3.262038974597697037181032587162, 4.20220546165416523216899732719, 4.96865286875585446550542106139, 6.057315996068188585856566051184, 7.02858228742366886980552200254, 7.54060716366090887202347102455, 8.79988196398454439619317415665, 9.85845540526403727803632828448, 10.62455994638038631486448785021, 11.71249455040415589822754600410, 12.2171644786043620475995320113, 12.73187557264863629381161370394, 13.50374356841181739324606518696, 14.44618550575002515434392551541, 15.09320062566603922295496812023, 16.28546981631025863385423569041, 16.98072525460987927421074750899, 17.88418676376270847831897513957, 18.991565605479448473591235122296, 19.48520823991923941542243083474, 19.999799137038112386579944221491, 20.589789759317740914383361088721

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