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Properties

Label 1-6017-6017.5389-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.617 - 0.786i$
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  + (0.623 + 0.781i)2-s + (0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.900 + 0.433i)5-s + (0.900 − 0.433i)6-s + (−0.222 − 0.974i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (0.900 + 0.433i)12-s + (−0.623 + 0.781i)13-s + (0.623 − 0.781i)14-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + (−0.900 − 0.433i)17-s + (−0.222 − 0.974i)18-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.900 + 0.433i)5-s + (0.900 − 0.433i)6-s + (−0.222 − 0.974i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (0.900 + 0.433i)12-s + (−0.623 + 0.781i)13-s + (0.623 − 0.781i)14-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + (−0.900 − 0.433i)17-s + (−0.222 − 0.974i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (5389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.617 - 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.721763642 - 0.8375689411i\)
\(L(\frac12)\) \(\approx\) \(1.721763642 - 0.8375689411i\)
\(L(1)\) \(\approx\) \(1.398895299 + 0.1401879550i\)
\(L(1)\) \(\approx\) \(1.398895299 + 0.1401879550i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 + (-0.623 + 0.781i)T \)
23 \( 1 + (-0.623 + 0.781i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (0.222 - 0.974i)T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.623 - 0.781i)T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−17.81368479067406504477988184058, −17.289924305524143514609232811323, −16.23442355069309582546157787539, −15.74945048029493185393153941302, −14.97750188039480523730942718711, −14.6337576621210071503482036789, −13.81873365898661730494326664498, −13.19011255317034829800193962548, −12.47053000934221426821905202640, −12.10068082847594924808372228899, −10.951145073580835277098985349561, −10.58729483391295884253621521599, −9.94319502937172698848637461534, −9.158385396035006741814018749149, −8.92333930772965465602593284036, −8.12235455818584532096099176292, −6.59125450393399582524595716829, −6.00460875414041945860912309195, −5.4457288806429364938248740505, −4.62694682168784989941555999360, −4.39702553700372390744723180224, −3.14518111391936357813181234174, −2.48932807709661609710945127460, −2.26469790060647387334892351909, −0.94926399870039046973890275884, 0.37456320398712761670832469271, 1.73106766328838737689887694447, 2.34313154717792147770381258496, 3.08024409889215166035712464886, 4.018543408912224099320721474055, 4.63435211382504722449599039387, 5.70869118048553311161531219261, 6.3212245356956811749050383025, 6.66667016683142312180626977109, 7.48805688360057196902515090907, 7.80831521365148206652972918088, 8.88177812836004855921561944652, 9.44083980079855850999525120720, 10.261023833301652170661794866895, 11.22171634072911246872124845518, 11.8912412309579728036626813638, 12.596930594449648642947168295507, 13.38558573712941033158399439642, 13.6800348584578077093616275888, 14.14679571569838186574120124625, 14.77504739729011384683904561200, 15.494124032114841005308881223172, 16.56563071174240969261323551911, 16.94304221180481773362313871579, 17.5544825344888963147809000192

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