Properties

Degree $1$
Conductor $6017$
Sign $0.299 + 0.954i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.315 − 0.948i)2-s + (−0.134 − 0.990i)3-s + (−0.800 + 0.598i)4-s + (−0.796 − 0.604i)5-s + (−0.897 + 0.440i)6-s + (−0.757 + 0.652i)7-s + (0.821 + 0.570i)8-s + (−0.963 + 0.266i)9-s + (−0.322 + 0.946i)10-s + (0.700 + 0.713i)12-s + (−0.753 + 0.657i)13-s + (0.858 + 0.512i)14-s + (−0.492 + 0.870i)15-s + (0.282 − 0.959i)16-s + (−0.503 + 0.863i)17-s + (0.556 + 0.830i)18-s + ⋯
L(s,χ)  = 1  + (−0.315 − 0.948i)2-s + (−0.134 − 0.990i)3-s + (−0.800 + 0.598i)4-s + (−0.796 − 0.604i)5-s + (−0.897 + 0.440i)6-s + (−0.757 + 0.652i)7-s + (0.821 + 0.570i)8-s + (−0.963 + 0.266i)9-s + (−0.322 + 0.946i)10-s + (0.700 + 0.713i)12-s + (−0.753 + 0.657i)13-s + (0.858 + 0.512i)14-s + (−0.492 + 0.870i)15-s + (0.282 − 0.959i)16-s + (−0.503 + 0.863i)17-s + (0.556 + 0.830i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.299 + 0.954i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.299 + 0.954i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.299 + 0.954i$
Motivic weight: \(0\)
Character: $\chi_{6017} (79, \cdot )$
Sato-Tate group: $\mu(910)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.299 + 0.954i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.1932621767 + 0.1418577854i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.1932621767 + 0.1418577854i\)
\(L(\chi,1)\) \(\approx\) \(0.4603260751 - 0.2773633352i\)
\(L(1,\chi)\) \(\approx\) \(0.4603260751 - 0.2773633352i\)

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

−17.40703092395335399440676453697, −16.79068639568938602700368755684, −16.08807254446821066711601229095, −15.702134179551044734833577957629, −15.23297778780644075675156812382, −14.53777521287611105040452245177, −13.81335337158973651882246529612, −13.36753600563461542553020918703, −12.0983386563499254704258387126, −11.61538501722821702749552839048, −10.62229294232557674061067500245, −10.13805963151451825409235567883, −9.69789925471683793481499630475, −8.971256867161910920170271081994, −7.9714069409765203933896421560, −7.62848852914520143764422982869, −6.75465010740882158032245736866, −6.22140570178794464024538019651, −5.36809986374131904463169962195, −4.47654769531120801829487671623, −4.1983848684619542150467382266, −3.17189860365243723777937185553, −2.65934240094624671614808186445, −0.75967849419278909023227556505, −0.11789790167920557842706688888, 0.92783568307903317475680007789, 1.72655258531695140939438118318, 2.472623968044340156237306188, 3.19701855160052071632160351223, 3.98250334866745918447452671252, 4.80547816738329851306312243302, 5.580372196115435836974678731944, 6.44610861005224196830550555094, 7.29627022999038675665201457847, 7.9168596953338464741125442363, 8.56136364431142555130629961849, 9.13308096876422374255191884142, 9.79627728044763546337351719764, 10.72282490810084086048814791136, 11.49536748964495674652843952730, 12.132636041353302803722813494311, 12.32289213916125403108749180560, 12.91851221226094318994784163444, 13.69345519612871295295032363012, 14.29784351199536046422024827011, 15.235121439793215182829827924784, 16.287604678999762340312325726207, 16.4564813078052287730342180299, 17.488483504309530721642428859590, 17.91661954062762705111194540959

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