Properties

Degree 1
Conductor 37
Sign $0.320 + 0.947i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + i·6-s + (−0.5 − 0.866i)7-s + i·8-s + (−0.5 + 0.866i)9-s + 10-s − 11-s + (−0.5 + 0.866i)12-s + (0.866 − 0.5i)13-s i·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + ⋯
L(s,χ)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + i·6-s + (−0.5 − 0.866i)7-s + i·8-s + (−0.5 + 0.866i)9-s + 10-s − 11-s + (−0.5 + 0.866i)12-s + (0.866 − 0.5i)13-s i·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.320 + 0.947i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.320 + 0.947i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(37\)
\( \varepsilon \)  =  $0.320 + 0.947i$
motivic weight  =  \(0\)
character  :  $\chi_{37} (8, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 37,\ (1:\ ),\ 0.320 + 0.947i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.214878642 + 1.588705544i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.214878642 + 1.588705544i\)
\(L(\chi,1)\)  \(\approx\)  \(1.788899329 + 0.8842390261i\)
\(L(1,\chi)\)  \(\approx\)  \(1.788899329 + 0.8842390261i\)

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

−35.01186838611360584385674063343, −33.70418417569581147698496796401, −32.44782608430526579586468002600, −31.169153920152052959432434091802, −30.56541542877876940899920486769, −28.981540388896056777444053422547, −28.77952340206667729225846412995, −26.15064172844240294351249134154, −25.185994030227197422839716216557, −24.07386467213252080654015074150, −22.74825255930772399367984767452, −21.53658318833374052599655366013, −20.39839012866768381106677004500, −18.84607377844038018197071116142, −18.20823105198513430425428023662, −15.735376192637147899707843847643, −14.34504630906104646115409071771, −13.342609028580464839831913035940, −12.35328010993638235746327216750, −10.658998884502228069219293882543, −8.99777412516335396985127923721, −6.7475769032492476719774631092, −5.6824733784314758163677025145, −3.10701424287336882713489175299, −1.937992876899654787260678737377, 2.879385196491709182319651624507, 4.484115760525524703752214971256, 5.82306433277190544103062121737, 7.79426069992086584105623131643, 9.45916157599402741819201088866, 10.95509100079841055132150098543, 13.33432963263049683323164711543, 13.63825153032010225657527703777, 15.47319045165383180818238903034, 16.30408133763966006073614798407, 17.61291363123304929658793610378, 20.17709483797286188836548309907, 20.838668062863841128854470877873, 22.0211830422638499961370973998, 23.206279951262925653279670161448, 24.71066482671482379292765836842, 25.85376882896272511826575381307, 26.5505104733957572481513935030, 28.4791402592070744765104733059, 29.72826054011695556284856289814, 31.16412993949204206322847061738, 32.21791630223314197995884199460, 33.109954340164283052452655188053, 33.65852692686942367308486846130, 35.5053932190607241685130514788

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