Properties

Degree $1$
Conductor $6017$
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

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.928 - 0.370i$
Motivic weight: \(0\)
Character: $\chi_{6017} (145, \cdot )$
Sato-Tate group: $\mu(390)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.928 - 0.370i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.6427301373 - 0.1235811979i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.6427301373 - 0.1235811979i\)
\(L(\chi,1)\) \(\approx\) \(0.9792488130 + 0.5564463378i\)
\(L(1,\chi)\) \(\approx\) \(0.9792488130 + 0.5564463378i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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