Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $0.928 + 0.370i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.997 − 0.0643i)2-s + (−0.309 − 0.951i)3-s + (0.991 − 0.128i)4-s + (−0.513 − 0.857i)5-s + (−0.369 − 0.929i)6-s + (−0.962 − 0.270i)7-s + (0.981 − 0.192i)8-s + (−0.809 + 0.587i)9-s + (−0.568 − 0.822i)10-s + (−0.428 − 0.903i)12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (−0.657 + 0.753i)15-s + (0.966 − 0.254i)16-s + (−0.581 − 0.813i)17-s + (−0.769 + 0.638i)18-s + ⋯
L(s,χ)  = 1  + (0.997 − 0.0643i)2-s + (−0.309 − 0.951i)3-s + (0.991 − 0.128i)4-s + (−0.513 − 0.857i)5-s + (−0.369 − 0.929i)6-s + (−0.962 − 0.270i)7-s + (0.981 − 0.192i)8-s + (−0.809 + 0.587i)9-s + (−0.568 − 0.822i)10-s + (−0.428 − 0.903i)12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (−0.657 + 0.753i)15-s + (0.966 − 0.254i)16-s + (−0.581 − 0.813i)17-s + (−0.769 + 0.638i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.928 + 0.370i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.928 + 0.370i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $0.928 + 0.370i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (83, \cdot )$
Sato-Tate  :  $\mu(390)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ 0.928 + 0.370i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6427301373 + 0.1235811979i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6427301373 + 0.1235811979i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9792488130 - 0.5564463378i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9792488130 - 0.5564463378i\)

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

−17.439697748600134480102529106781, −16.702675994299602881444992442088, −16.20319489775035563971164130449, −15.51673860564002391994493864049, −15.13453753426346934512461622985, −14.51075775349923788501336290310, −14.00505226627462793758362168940, −12.91624682642908771060770320267, −12.425581277860400601433891721316, −11.76689088273284726861490200496, −11.08186165415087148498079149018, −10.46600944462534120451583075149, −10.07520877134144908892429769572, −9.02702242152884395227298511595, −8.36549730401732447177448747436, −7.186606307099685323112138825040, −6.66042916754262913672497383470, −6.25093620350830440230953972146, −5.3223752157154224068199068211, −4.58215494498515942440737384985, −4.02605040645090474962744419328, −3.27607163265671743226246659003, −2.76804217574358369398839794659, −2.00335902943633200725250590245, −0.13891571097835614864816101607, 0.78100889897749004571919359490, 1.78061332833321028860481733503, 2.49826622926965333515047661652, 3.33340053851936709587107109548, 4.09556847001090001565800882389, 4.84821629581736959060901372693, 5.62320283129478904996155697472, 6.06903887195958349533105937510, 7.05164233433305596476482235533, 7.4135468705011382785058765234, 8.07138105423364191445948152216, 9.062705049734078016969925133319, 9.895234067812117049779290786831, 10.688272080770270942560420791262, 11.52186984871494046376398577946, 12.06887731335266571071923024124, 12.56784888516469457075490990373, 12.99697742597032292682105169379, 13.71129895462952452392880253803, 14.14226995538024079914489708993, 15.2678381174027682786698636240, 15.68721376587346991127693412555, 16.459662737456705035137121531912, 16.96987702377062027840196343241, 17.493251523858553336983532230208

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