Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $0.791 - 0.610i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.341 + 0.939i)2-s + (0.0448 − 0.998i)3-s + (−0.766 − 0.642i)4-s + (0.405 + 0.914i)5-s + (0.923 + 0.383i)6-s + (−0.503 − 0.863i)7-s + (0.865 − 0.500i)8-s + (−0.995 − 0.0896i)9-s + (−0.997 + 0.0689i)10-s + (−0.675 + 0.736i)12-s + (0.691 − 0.722i)13-s + (0.983 − 0.178i)14-s + (0.931 − 0.364i)15-s + (0.175 + 0.984i)16-s + (−0.982 − 0.185i)17-s + (0.424 − 0.905i)18-s + ⋯
L(s,χ)  = 1  + (−0.341 + 0.939i)2-s + (0.0448 − 0.998i)3-s + (−0.766 − 0.642i)4-s + (0.405 + 0.914i)5-s + (0.923 + 0.383i)6-s + (−0.503 − 0.863i)7-s + (0.865 − 0.500i)8-s + (−0.995 − 0.0896i)9-s + (−0.997 + 0.0689i)10-s + (−0.675 + 0.736i)12-s + (0.691 − 0.722i)13-s + (0.983 − 0.178i)14-s + (0.931 − 0.364i)15-s + (0.175 + 0.984i)16-s + (−0.982 − 0.185i)17-s + (0.424 − 0.905i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.791 - 0.610i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.791 - 0.610i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $0.791 - 0.610i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (8, \cdot )$
Sato-Tate  :  $\mu(910)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ 0.791 - 0.610i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9643132106 - 0.3287277998i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9643132106 - 0.3287277998i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7895779063 + 0.05327836549i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7895779063 + 0.05327836549i\)

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

−17.786502278994589166269031062458, −17.21832139912318319617614717106, −16.39630869807405324260898076798, −15.95632461781301023396242487125, −15.449904968357995396967929671486, −14.31078730376072791920988513352, −13.70855270241609490058185264942, −13.138580197238690615720226943230, −12.35684896476821188630599293680, −11.77105942805825329769190524299, −11.138449706525955518565354061742, −10.48479413645193651945794025886, −9.57344787985780071965466540509, −9.28658359658213669466037234696, −8.722738791060588455152281999598, −8.33581418673666706457845482161, −7.05044246158053774234045007684, −6.013309490824696109845485264876, −5.286641620109818358984454807982, −4.79299007372809964097618672591, −3.850332796919761654378210730417, −3.484173893871621138451789018813, −2.273974705000139538374590746377, −1.99816062720246638993270534724, −0.69345840983842664063111942147, 0.41897372644410233035421944745, 1.32718890058580704012735473635, 2.17118606768238671271130630886, 3.200227332594921910592470330921, 3.8316046157276410879833660266, 4.92735201641089085695416203081, 5.93712570014214466237935990289, 6.27479368015677296251659810641, 6.827460226898926677765214613030, 7.496184886819854276103202581752, 8.0475905913182521307873535297, 8.72996636560429365369737092942, 9.68581542374250338979978488355, 10.269554575963492218113016035776, 10.83223314976496301032193940742, 11.65543323055603260258002683129, 12.70431018925775783677964867433, 13.33212459684330237481391147953, 13.81825098746171426209577175769, 14.18161492246984412891806404587, 15.05735226401919540541937153252, 15.62767759773129028999583885528, 16.5431882575033791841798973514, 17.001998903243205062596267691329, 17.793271558387917613046981663893

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