Properties

Label 1-6017-6017.912-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.933 + 0.359i$
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  + (−0.222 − 0.974i)2-s + (0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.623 + 0.781i)5-s + (−0.623 − 0.781i)6-s + (−0.900 − 0.433i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (0.900 + 0.433i)10-s + (−0.623 + 0.781i)12-s + (0.222 − 0.974i)13-s + (−0.222 + 0.974i)14-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + (0.623 − 0.781i)17-s + (−0.900 − 0.433i)18-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.623 + 0.781i)5-s + (−0.623 − 0.781i)6-s + (−0.900 − 0.433i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (0.900 + 0.433i)10-s + (−0.623 + 0.781i)12-s + (0.222 − 0.974i)13-s + (−0.222 + 0.974i)14-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + (0.623 − 0.781i)17-s + (−0.900 − 0.433i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.933 + 0.359i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (912, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.933 + 0.359i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2496760457 - 1.341362685i\)
\(L(\frac12)\) \(\approx\) \(-0.2496760457 - 1.341362685i\)
\(L(1)\) \(\approx\) \(0.7030303445 - 0.6895192432i\)
\(L(1)\) \(\approx\) \(0.7030303445 - 0.6895192432i\)

Euler product

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

−18.140631801857037563934072123879, −17.068611864814363873480426365042, −16.469572330547419602577419501777, −16.16078260151445613093743685578, −15.536732925062348332527801240228, −14.99122736908134363241193416671, −14.26323167571418724788176690318, −13.6969270752902984518997997430, −12.875796840902777151985781149604, −12.50111748653542737905062652342, −11.53897984649989225828880152409, −10.41719439369273311350854085489, −9.84385666622824484329773086198, −9.06238358729545658702723671542, −8.8962485983700556405071512105, −8.04757825107948603836276551281, −7.52277185147961076632721496663, −6.77311691485048163482045865646, −5.83779905947857521322408003436, −5.25561538734861271418617419127, −4.38165973406404726450531240978, −3.71024607315782949315522180727, −3.31943346155757743984742668479, −1.878047207758235790097581371516, −1.17534674075575791757893753023, 0.41693289656532686410421024577, 0.95277704449486161622351292489, 2.29656097982520027311307751126, 2.90266394029405832105955623284, 3.21127936582102972490098912505, 3.98093767701786357030555906294, 4.68471017365315711233341677718, 5.926128354172440979482642466104, 6.83015628490930535021823595790, 7.43509512748378473334685548379, 7.98973013049926395098722876013, 8.70306540551577613975643559789, 9.51037801941092590434902991826, 10.035654560010103283504026564623, 10.63097532489047151056077380124, 11.40655854748253409016898287175, 12.15400246537943397409950842501, 12.69797639896370859950640039292, 13.423261445198250946805918544052, 13.84337906655020812227988552873, 14.53176752635255736729555469216, 15.42865380375238951267901172663, 15.78234936539583806337414817709, 16.90984464037128443812798728222, 17.54223306819790067886555771191

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