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Properties

Label 1-6017-6017.1025-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.662 + 0.749i$
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  + (0.883 − 0.468i)2-s + (−0.858 − 0.512i)3-s + (0.560 − 0.828i)4-s + (−0.810 − 0.585i)5-s + (−0.998 − 0.0506i)6-s + (−0.814 + 0.580i)7-s + (0.106 − 0.994i)8-s + (0.473 + 0.880i)9-s + (−0.990 − 0.137i)10-s + (−0.905 + 0.423i)12-s + (−0.251 + 0.967i)13-s + (−0.447 + 0.894i)14-s + (0.395 + 0.918i)15-s + (−0.371 − 0.928i)16-s + (0.893 + 0.448i)17-s + (0.831 + 0.555i)18-s + ⋯
L(s)  = 1  + (0.883 − 0.468i)2-s + (−0.858 − 0.512i)3-s + (0.560 − 0.828i)4-s + (−0.810 − 0.585i)5-s + (−0.998 − 0.0506i)6-s + (−0.814 + 0.580i)7-s + (0.106 − 0.994i)8-s + (0.473 + 0.880i)9-s + (−0.990 − 0.137i)10-s + (−0.905 + 0.423i)12-s + (−0.251 + 0.967i)13-s + (−0.447 + 0.894i)14-s + (0.395 + 0.918i)15-s + (−0.371 − 0.928i)16-s + (0.893 + 0.448i)17-s + (0.831 + 0.555i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.662 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1323357323 - 0.2935079686i\)
\(L(\frac12)\) \(\approx\) \(-0.1323357323 - 0.2935079686i\)
\(L(1)\) \(\approx\) \(0.8223793778 - 0.4418416285i\)
\(L(1)\) \(\approx\) \(0.8223793778 - 0.4418416285i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.883 - 0.468i)T \)
3 \( 1 + (-0.858 - 0.512i)T \)
5 \( 1 + (-0.810 - 0.585i)T \)
7 \( 1 + (-0.814 + 0.580i)T \)
13 \( 1 + (-0.251 + 0.967i)T \)
17 \( 1 + (0.893 + 0.448i)T \)
19 \( 1 + (-0.927 + 0.372i)T \)
23 \( 1 + (0.944 - 0.327i)T \)
29 \( 1 + (0.276 + 0.961i)T \)
31 \( 1 + (0.127 - 0.991i)T \)
37 \( 1 + (-0.416 - 0.909i)T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.0632 - 0.997i)T \)
47 \( 1 + (-0.962 - 0.270i)T \)
53 \( 1 + (-0.0126 + 0.999i)T \)
59 \( 1 + (-0.870 + 0.493i)T \)
61 \( 1 + (0.583 + 0.812i)T \)
67 \( 1 + (-0.819 - 0.572i)T \)
71 \( 1 + (-0.876 + 0.480i)T \)
73 \( 1 + (-0.220 - 0.975i)T \)
79 \( 1 + (0.999 + 0.0414i)T \)
83 \( 1 + (0.339 + 0.940i)T \)
89 \( 1 + (0.868 + 0.495i)T \)
97 \( 1 + (0.782 - 0.622i)T \)
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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

−17.709845590122348432698443126182, −17.37092742727729511608004071283, −16.61780082705333504441821619556, −15.97073744206497999426994657635, −15.65341469012182738859926790403, −14.89129663142021249161238600717, −14.480261318408343190623502347512, −13.42746101419744614507989941449, −12.83125440930162418584159389504, −12.253022652852408286542275044490, −11.619880752966471276008079114145, −10.92944275913145412804698681666, −10.40570872790601483930367813447, −9.735012614478367750565301526, −8.643861360260570494956926634670, −7.72064515224138480190314342867, −7.23398889233207046038222340421, −6.466347649586851371073597029264, −6.12107779205111792086536076351, −4.995474961231242090602336092959, −4.70213073283826556830304022778, −3.65023403181582179752499695317, −3.34569973458475501197162357244, −2.6279074708227189624887654708, −0.99613389721467709467693354270, 0.079858137373722695125024533192, 1.131679511494316443080871403877, 1.87172538226516231108265212360, 2.75831162832738303747985932395, 3.65848140157867814297261990528, 4.34799550989793998626428791364, 4.98137916801339330290993745493, 5.731652702142578882627019097726, 6.29436969789676851748059615556, 7.00025528247799006642430978251, 7.624462072228025487657201803703, 8.69180719724333881072668608527, 9.37647782439389618786903392402, 10.292387859933895858523099634806, 10.89254743593342332490607510390, 11.61503614757288160500890556976, 12.22350386669866772208244392873, 12.51287608651108473260403963797, 13.0498472825821610067795064686, 13.79319315555870877467481486414, 14.79727160906041475967462786849, 15.19255450674718131913960591771, 16.16351292305406524820589553248, 16.5116631809460466398229983689, 16.952990853580455306668096849816

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