Properties

Degree $1$
Conductor $6017$
Sign $-0.961 - 0.275i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.993 − 0.112i)2-s + (0.809 − 0.587i)3-s + (0.974 + 0.223i)4-s + (0.231 + 0.972i)5-s + (−0.870 + 0.493i)6-s + (−0.953 − 0.301i)7-s + (−0.943 − 0.331i)8-s + (0.309 − 0.951i)9-s + (−0.120 − 0.992i)10-s + (0.919 − 0.391i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (0.759 + 0.650i)15-s + (0.899 + 0.435i)16-s + (−0.769 − 0.638i)17-s + (−0.414 + 0.910i)18-s + ⋯
L(s,χ)  = 1  + (−0.993 − 0.112i)2-s + (0.809 − 0.587i)3-s + (0.974 + 0.223i)4-s + (0.231 + 0.972i)5-s + (−0.870 + 0.493i)6-s + (−0.953 − 0.301i)7-s + (−0.943 − 0.331i)8-s + (0.309 − 0.951i)9-s + (−0.120 − 0.992i)10-s + (0.919 − 0.391i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (0.759 + 0.650i)15-s + (0.899 + 0.435i)16-s + (−0.769 − 0.638i)17-s + (−0.414 + 0.910i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.961 - 0.275i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.961 - 0.275i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.961 - 0.275i$
Motivic weight: \(0\)
Character: $\chi_{6017} (1460, \cdot )$
Sato-Tate group: $\mu(390)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.961 - 0.275i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.09740147166 - 0.6922194437i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.09740147166 - 0.6922194437i\)
\(L(\chi,1)\) \(\approx\) \(0.7251205607 - 0.2314018232i\)
\(L(1,\chi)\) \(\approx\) \(0.7251205607 - 0.2314018232i\)

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

−17.934857701800745435816379135280, −17.19069630034510395282717725468, −16.655916329765047975531965197295, −16.11981510147650033702049139673, −15.540776830583367612578150095810, −15.06668120973553317458243329422, −14.1738795659445978209407768025, −13.317087509265452160408244764362, −12.91256347956395526950201485956, −11.86001132450025900227228213855, −11.444989036993499963032033200743, −10.24628191295603670842318446521, −9.86365240741803799931839279581, −9.30814128259193239932192426687, −8.80422617436094393742678242420, −8.31079240615917374987683609461, −7.34691643509647880853897934569, −6.82950366897419316783072944682, −5.81280459971394807327972966862, −5.179131255455612341447601942063, −4.29448570985811428206616343515, −3.41535806932620252735231306327, −2.647641525747596267854768508304, −1.94872679522671714370215717509, −1.15159690322739467430802241462, 0.229584607194405246242131725226, 1.09480144018938251352602798938, 2.33676518050262537663340087596, 2.58628653643152939511452843873, 3.28101144970196160147404434228, 3.9605014808657792681758982892, 5.499720349277951464370810397156, 6.26491059321581833440198594624, 7.02283293344606704264394848970, 7.25321378155033042645688735436, 7.900522784858591962244384893776, 8.884069070195264795540410479310, 9.44029077861885590590082276922, 9.947526931974690828169866011132, 10.58164838774729413046693614816, 11.36354985977139173168584343256, 12.088605055132760303706303090161, 12.95463935332755234591145823431, 13.27308242085783612785452504109, 14.40409774089800822723360965182, 14.66412531110174622976817984777, 15.65789595690034452161753531927, 15.93696696853154450381320784892, 16.98748738885775478487198662229, 17.64765939559487219612029594882

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