Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $0.936 - 0.350i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.779 + 0.626i)2-s + (−0.809 + 0.587i)3-s + (0.215 + 0.976i)4-s + (−0.168 + 0.985i)5-s + (−0.998 − 0.0483i)6-s + (−0.262 + 0.964i)7-s + (−0.443 + 0.896i)8-s + (0.309 − 0.951i)9-s + (−0.748 + 0.663i)10-s + (−0.748 − 0.663i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (−0.443 − 0.896i)15-s + (−0.906 + 0.421i)16-s + (−0.527 − 0.849i)17-s + (0.836 − 0.548i)18-s + ⋯
L(s,χ)  = 1  + (0.779 + 0.626i)2-s + (−0.809 + 0.587i)3-s + (0.215 + 0.976i)4-s + (−0.168 + 0.985i)5-s + (−0.998 − 0.0483i)6-s + (−0.262 + 0.964i)7-s + (−0.443 + 0.896i)8-s + (0.309 − 0.951i)9-s + (−0.748 + 0.663i)10-s + (−0.748 − 0.663i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (−0.443 − 0.896i)15-s + (−0.906 + 0.421i)16-s + (−0.527 − 0.849i)17-s + (0.836 − 0.548i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.936 - 0.350i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.936 - 0.350i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $0.936 - 0.350i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (3028, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ 0.936 - 0.350i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.1195093030 + 0.02166186863i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.1195093030 + 0.02166186863i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5951491378 + 0.7598154182i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5951491378 + 0.7598154182i\)

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

−16.92973181967667980613026588539, −16.6039505422323164252503095332, −15.80432919051263358175154752109, −15.187199892909566761125883096944, −14.11127803734502168004879031441, −13.39264536061599444306054908253, −13.270333538486266723752728456849, −12.54672618418948949637617576133, −11.75053583342316847134570132265, −11.457306177806689679605659971124, −10.59121120326285703256196384131, −10.128593265032904110839501247572, −9.132533454365490164140828575074, −8.50261558537103347329458347291, −7.33958705233981956186259413406, −6.857575252868676573723483322534, −6.16426475629206561732184847407, −5.43471699354101917021636334136, −4.57740962693549298939869298190, −4.33478355511274456578121635954, −3.46439805341358242708660596401, −2.28863815197176272754962664708, −1.53630570956480850681180035945, −0.931384652837750300675484309402, −0.02844452868446660365508799295, 1.69625647131659747935626143623, 2.935425736450161479708385537, 3.21898819542868064989502999149, 3.94744374824414480135209127425, 5.06814917309708554190301763362, 5.36992316453620722790810414117, 6.12128884940269576956426796861, 6.658885941895243569468948021571, 7.34380594794304155820274211166, 8.154393785345918162260029672916, 8.99305538034305593489122251019, 9.787306837742510844044403596, 10.48698105589182394570797302529, 11.37840192353294263306490119896, 11.6621325183059824988828117432, 12.34847980026727030597030193589, 13.09306810385638307343894903236, 13.83033005098368995174297675687, 14.62446483320558455353902124462, 15.35633284308550863474831390845, 15.482283827977392384394516929877, 16.09661022831800584140065193267, 16.91631953203314319078379888316, 17.590269176847707369884741365354, 18.2627671049438377042327947882

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