Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $0.994 - 0.106i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.295 + 0.955i)2-s + (0.550 − 0.834i)3-s + (−0.824 + 0.565i)4-s + (−0.348 − 0.937i)5-s + (0.960 + 0.279i)6-s + (−0.894 + 0.446i)7-s + (−0.783 − 0.620i)8-s + (−0.393 − 0.919i)9-s + (0.792 − 0.609i)10-s + (0.0172 + 0.999i)12-s + (−0.473 + 0.880i)13-s + (−0.691 − 0.722i)14-s + (−0.974 − 0.225i)15-s + (0.361 − 0.932i)16-s + (−0.113 − 0.993i)17-s + (0.762 − 0.647i)18-s + ⋯
L(s,χ)  = 1  + (0.295 + 0.955i)2-s + (0.550 − 0.834i)3-s + (−0.824 + 0.565i)4-s + (−0.348 − 0.937i)5-s + (0.960 + 0.279i)6-s + (−0.894 + 0.446i)7-s + (−0.783 − 0.620i)8-s + (−0.393 − 0.919i)9-s + (0.792 − 0.609i)10-s + (0.0172 + 0.999i)12-s + (−0.473 + 0.880i)13-s + (−0.691 − 0.722i)14-s + (−0.974 − 0.225i)15-s + (0.361 − 0.932i)16-s + (−0.113 − 0.993i)17-s + (0.762 − 0.647i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.994 - 0.106i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.994 - 0.106i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $0.994 - 0.106i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (105, \cdot )$
Sato-Tate  :  $\mu(910)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ 0.994 - 0.106i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.186952109 - 0.06330547682i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.186952109 - 0.06330547682i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9568232369 + 0.1240818604i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9568232369 + 0.1240818604i\)

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

−17.61592095045315365340946171204, −17.361803270849927510214790773179, −16.15224434424447906834749514623, −15.58118162034250447876415688439, −14.965860301009165184030961442529, −14.53066272955157611561358058458, −13.6116999694428113725927566817, −13.321813585873452387510343502798, −12.46505169511092061219334200443, −11.61703773436969297682092560821, −10.97712990041643556726601512752, −10.38749782688019445279817341211, −9.901251455767996443636567224891, −9.48980841471694062847411304350, −8.42951387337221159777522302272, −7.88695909545022415530537393187, −6.93961018312334519794514172280, −6.04088824619721979198795525649, −5.36463845815986156110213068015, −4.34886278979065246911475752688, −3.84311829463012445913795755467, −3.27264608518818442613313877146, −2.65500295916619068698308005144, −2.043484459793192366354553282097, −0.54702411994153431195362439225, 0.435993255469143647436960252895, 1.50586474136438334966461451582, 2.556159023279097616679170370827, 3.29467384349280513394384035170, 4.11017446082572863660058054863, 4.746615084735242504211681248900, 5.77049058156834130010160920085, 6.173584716392473439309580607230, 7.01481476781792717945306161377, 7.59508668763503947771164860359, 8.2141163429114289124467188970, 8.8923771677792374934914013854, 9.456817827614291436604761928173, 9.88382040422908701388775363188, 11.7579510859775905800818767543, 11.90743469640786582300698004785, 12.57435728331037235608724608553, 13.37276538343358010269520439167, 13.52090688906958875536051784095, 14.61991973568170720345495690802, 14.85551130043467438833927779537, 15.95565142604795741778882812404, 16.33278166903987461567643107784, 16.73945812989599818130946052898, 17.73359146992303375672296182999

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