Properties

Degree $1$
Conductor $6017$
Sign $0.835 - 0.548i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.443 + 0.896i)2-s + (0.309 + 0.951i)3-s + (−0.607 − 0.794i)4-s + (0.485 − 0.873i)5-s + (−0.989 − 0.144i)6-s + (0.715 − 0.698i)7-s + (0.981 − 0.192i)8-s + (−0.809 + 0.587i)9-s + (0.568 + 0.822i)10-s + (0.568 − 0.822i)12-s + (−0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (0.981 + 0.192i)15-s + (−0.262 + 0.964i)16-s + (0.995 − 0.0965i)17-s + (−0.168 − 0.985i)18-s + ⋯
L(s,χ)  = 1  + (−0.443 + 0.896i)2-s + (0.309 + 0.951i)3-s + (−0.607 − 0.794i)4-s + (0.485 − 0.873i)5-s + (−0.989 − 0.144i)6-s + (0.715 − 0.698i)7-s + (0.981 − 0.192i)8-s + (−0.809 + 0.587i)9-s + (0.568 + 0.822i)10-s + (0.568 − 0.822i)12-s + (−0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (0.981 + 0.192i)15-s + (−0.262 + 0.964i)16-s + (0.995 − 0.0965i)17-s + (−0.168 − 0.985i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.835 - 0.548i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.835 - 0.548i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.835 - 0.548i$
Motivic weight: \(0\)
Character: $\chi_{6017} (2150, \cdot )$
Sato-Tate group: $\mu(65)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.835 - 0.548i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.8875093374 - 0.2652851434i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.8875093374 - 0.2652851434i\)
\(L(\chi,1)\) \(\approx\) \(0.8107050170 + 0.3364152791i\)
\(L(1,\chi)\) \(\approx\) \(0.8107050170 + 0.3364152791i\)

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

−17.92877446567596269373109665919, −17.31502673112729874461917380683, −17.089647729686950602920018122017, −15.7045810524190994244687377713, −14.85819039634102185839340653803, −14.30270425566056642110141453757, −13.908022163565795733896186331255, −12.93979970224897239058247390225, −12.42811838660648290780292802855, −11.914927367835612013302585131624, −11.187561859878199972449456076694, −10.55242223337182515107157170097, −9.81211669193775642608051045320, −9.18870051695763917049783081684, −8.37284118906844584829107316473, −7.658625507069826713358594536772, −7.48919139874555301058912370460, −6.15978679472991728643159043098, −5.808594063406655533280788176441, −4.72669722343362087081210916474, −3.75286948212292565831139514489, −2.88801718367093029843635446366, −2.32002979521702051593992802116, −1.95532225219456335885293754424, −0.97766398034208781186103000665, 0.270678357970581945766405707913, 1.492804431844828601660059233988, 2.04716640740485281678658369649, 3.4298218260332481944052783730, 4.33034806897662364807110081023, 4.76674548769773775642075796787, 5.26996820692822034154465876698, 6.0752955873212955055695073681, 6.930232731057898605526029382229, 7.941699840076257820153660968571, 8.16296790938864825412477158257, 9.03330494195915326455290810835, 9.526666247048262530086558441886, 10.23854512090962340534449839376, 10.58279838921503495040786276113, 11.63243760891932928311008841826, 12.42349282820225920588760699152, 13.4448082965300376937840397280, 14.015483081313594757476204360929, 14.401926273807359622959073279820, 15.062191073147316123619777490631, 15.80098068352436290611220190128, 16.554034327424730034317840142830, 16.873141164403068093218526412508, 17.291209890509909645069521082372

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