Properties

Degree $1$
Conductor $6017$
Sign $-0.895 + 0.445i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.854 − 0.518i)2-s + (0.550 − 0.834i)3-s + (0.461 + 0.887i)4-s + (−0.969 − 0.246i)5-s + (−0.903 + 0.427i)6-s + (0.965 + 0.259i)7-s + (0.0655 − 0.997i)8-s + (−0.393 − 0.919i)9-s + (0.700 + 0.713i)10-s + (0.994 + 0.103i)12-s + (−0.473 + 0.880i)13-s + (−0.691 − 0.722i)14-s + (−0.739 + 0.673i)15-s + (−0.573 + 0.819i)16-s + (0.361 − 0.932i)17-s + (−0.141 + 0.989i)18-s + ⋯
L(s,χ)  = 1  + (−0.854 − 0.518i)2-s + (0.550 − 0.834i)3-s + (0.461 + 0.887i)4-s + (−0.969 − 0.246i)5-s + (−0.903 + 0.427i)6-s + (0.965 + 0.259i)7-s + (0.0655 − 0.997i)8-s + (−0.393 − 0.919i)9-s + (0.700 + 0.713i)10-s + (0.994 + 0.103i)12-s + (−0.473 + 0.880i)13-s + (−0.691 − 0.722i)14-s + (−0.739 + 0.673i)15-s + (−0.573 + 0.819i)16-s + (0.361 − 0.932i)17-s + (−0.141 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.895 + 0.445i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.895 + 0.445i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.895 + 0.445i$
Motivic weight: \(0\)
Character: $\chi_{6017} (270, \cdot )$
Sato-Tate group: $\mu(910)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.895 + 0.445i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.1335959023 - 0.5680032746i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.1335959023 - 0.5680032746i\)
\(L(\chi,1)\) \(\approx\) \(0.5942426805 - 0.3820022679i\)
\(L(1,\chi)\) \(\approx\) \(0.5942426805 - 0.3820022679i\)

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

−18.11851012799243788551092008987, −17.234170717161818787964493642744, −16.7556631189669702299695522429, −16.14054993194932888097709222034, −15.43718355206322888758788670055, −14.79803427884158667099273522672, −14.61567808891957902041229418369, −14.00559577670683407166261145911, −12.67365581769515873933164760110, −12.07555523950743475446402994668, −10.97212491173646694359705716852, −10.59622486042266079587483230136, −10.41749740987823288890232367258, −9.22919944845267721932176353819, −8.60731674899434722436665936896, −8.13978217993737343348801114133, −7.54627376337677790448054078571, −7.064644299985854640117606523101, −5.71214182662893037876209086321, −5.35838479864771247795260480774, −4.27687079045132364579455923720, −3.90075960395095141483512061105, −2.79777081071502590943906865062, −2.07401023078774670473668673389, −1.05353636351809976754631640367, 0.20524860602809860690528411375, 1.15270828171120515734411313025, 1.85381366734647499429200038602, 2.57991404112648628357096900825, 3.29294727902426927416399988613, 4.18611231607713101070596715306, 4.82804409246568278827230050306, 6.019070239043502749531452894633, 7.10726949505726946538475592656, 7.38180979350490726883708763596, 7.94865033139097443406022499476, 8.69311495033223990133836412348, 9.13354311573584951051019661700, 9.78534123591136020027999297290, 11.063487434926782738457991881457, 11.44541917495490709969023125754, 11.91152240636524620124516874715, 12.47449945739617356435135651798, 13.310539681314878391139452870630, 13.93235262939604538035837179302, 14.90108181446743665246924901032, 15.281116299298100313017843600329, 16.1338728715637740281316144458, 16.95681368104383604179314812236, 17.4009107474239219304097739164

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