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Properties

Label 1-6017-6017.1976-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.652 + 0.758i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.652 + 0.758i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1976, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.652 + 0.758i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.584 - 0.811i)T \)
3 \( 1 + (-0.753 + 0.657i)T \)
5 \( 1 + (0.947 - 0.318i)T \)
7 \( 1 + (0.348 - 0.937i)T \)
13 \( 1 + (-0.936 - 0.351i)T \)
17 \( 1 + (0.498 - 0.867i)T \)
19 \( 1 + (-0.147 - 0.989i)T \)
23 \( 1 + (-0.997 - 0.0689i)T \)
29 \( 1 + (0.455 + 0.890i)T \)
31 \( 1 + (0.209 + 0.977i)T \)
37 \( 1 + (-0.411 + 0.911i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.479 + 0.877i)T \)
47 \( 1 + (-0.943 + 0.331i)T \)
53 \( 1 + (0.665 + 0.746i)T \)
59 \( 1 + (0.681 + 0.732i)T \)
61 \( 1 + (-0.430 - 0.902i)T \)
67 \( 1 + (0.386 + 0.922i)T \)
71 \( 1 + (-0.634 - 0.773i)T \)
73 \( 1 + (0.977 - 0.212i)T \)
79 \( 1 + (-0.982 - 0.185i)T \)
83 \( 1 + (0.715 - 0.698i)T \)
89 \( 1 + (-0.725 - 0.688i)T \)
97 \( 1 + (-0.993 - 0.117i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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