Properties

Degree $1$
Conductor $6017$
Sign $-0.652 - 0.758i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.584 + 0.811i)2-s + (−0.753 − 0.657i)3-s + (−0.315 − 0.948i)4-s + (0.947 + 0.318i)5-s + (0.974 − 0.225i)6-s + (0.348 + 0.937i)7-s + (0.954 + 0.299i)8-s + (0.134 + 0.990i)9-s + (−0.813 + 0.582i)10-s + (−0.386 + 0.922i)12-s + (−0.936 + 0.351i)13-s + (−0.963 − 0.266i)14-s + (−0.503 − 0.863i)15-s + (−0.800 + 0.598i)16-s + (0.498 + 0.867i)17-s + (−0.882 − 0.470i)18-s + ⋯
L(s,χ)  = 1  + (−0.584 + 0.811i)2-s + (−0.753 − 0.657i)3-s + (−0.315 − 0.948i)4-s + (0.947 + 0.318i)5-s + (0.974 − 0.225i)6-s + (0.348 + 0.937i)7-s + (0.954 + 0.299i)8-s + (0.134 + 0.990i)9-s + (−0.813 + 0.582i)10-s + (−0.386 + 0.922i)12-s + (−0.936 + 0.351i)13-s + (−0.963 − 0.266i)14-s + (−0.503 − 0.863i)15-s + (−0.800 + 0.598i)16-s + (0.498 + 0.867i)17-s + (−0.882 − 0.470i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.652 - 0.758i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.652 - 0.758i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.652 - 0.758i$
Motivic weight: \(0\)
Character: $\chi_{6017} (338, \cdot )$
Sato-Tate group: $\mu(910)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.652 - 0.758i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.08880699506 + 0.1934975803i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.08880699506 + 0.1934975803i\)
\(L(\chi,1)\) \(\approx\) \(0.5765044782 + 0.2476234164i\)
\(L(1,\chi)\) \(\approx\) \(0.5765044782 + 0.2476234164i\)

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

−17.41342261355990326170735165517, −16.73722693040589514742271503781, −16.4185551804862290656195591747, −15.576300531330870314356193748038, −14.49227730384673369381521548413, −13.9364234882787576402423325728, −13.23964994046161838976993081889, −12.466601798188354442415496038301, −11.9197405049134236966799299008, −11.24107712470344135201946212345, −10.3828011972530653398915846201, −10.18229151380172270922612174169, −9.5900710105039509991086421673, −8.87254080980763881013699053596, −8.10699754167202100945443262032, −7.08202150942071059698357840519, −6.69653535571582044387235168164, −5.46670315372799427320480603667, −4.794924755815110990829801907166, −4.512025901929902638468905461821, −3.343853714085311728239581831628, −2.77528445706974716073004156931, −1.64960022858244066126385197917, −1.00173429484952889604968329073, −0.079396616753867307918396286961, 1.29186728006941727186886949878, 2.02973705089124570439014080450, 2.32521097089967292744406275883, 3.94856973359911242372432444374, 5.0305887882878245598420556609, 5.475288825815443324767598124126, 6.08834057369254407420869106371, 6.50699846554151098094696123130, 7.336141193485901316286992093017, 8.11548415460448760188614014401, 8.50784508962089694685941149976, 9.76187661395249582533744728948, 9.897311649775046780203291719010, 10.691039598979187512242213092810, 11.57719522277094815620949263400, 12.155791054915864734058000971306, 12.90383314982843637102736399316, 13.69514188506372052031552752550, 14.309471996734431952034408680736, 14.85544686557559112046967418927, 15.524968724463943922569977715080, 16.58488237145987624453627752247, 16.796977942470664071190487610073, 17.585320172390307418542832430258, 17.95300332655131217814548258016

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