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Properties

Label 1-6017-6017.1009-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.780 - 0.625i$
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  + (−0.824 + 0.565i)2-s + (0.393 + 0.919i)3-s + (0.361 − 0.932i)4-s + (0.757 − 0.652i)5-s + (−0.843 − 0.536i)6-s + (0.601 − 0.798i)7-s + (0.229 + 0.973i)8-s + (−0.691 + 0.722i)9-s + (−0.256 + 0.966i)10-s + (0.999 − 0.0345i)12-s + (0.550 + 0.834i)13-s + (−0.0448 + 0.998i)14-s + (0.897 + 0.440i)15-s + (−0.739 − 0.673i)16-s + (−0.974 + 0.225i)17-s + (0.161 − 0.986i)18-s + ⋯
L(s)  = 1  + (−0.824 + 0.565i)2-s + (0.393 + 0.919i)3-s + (0.361 − 0.932i)4-s + (0.757 − 0.652i)5-s + (−0.843 − 0.536i)6-s + (0.601 − 0.798i)7-s + (0.229 + 0.973i)8-s + (−0.691 + 0.722i)9-s + (−0.256 + 0.966i)10-s + (0.999 − 0.0345i)12-s + (0.550 + 0.834i)13-s + (−0.0448 + 0.998i)14-s + (0.897 + 0.440i)15-s + (−0.739 − 0.673i)16-s + (−0.974 + 0.225i)17-s + (0.161 − 0.986i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.780 - 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06708924183 + 0.1908391939i\)
\(L(\frac12)\) \(\approx\) \(-0.06708924183 + 0.1908391939i\)
\(L(1)\) \(\approx\) \(0.7477040506 + 0.2983944640i\)
\(L(1)\) \(\approx\) \(0.7477040506 + 0.2983944640i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.824 + 0.565i)T \)
3 \( 1 + (0.393 + 0.919i)T \)
5 \( 1 + (0.757 - 0.652i)T \)
7 \( 1 + (0.601 - 0.798i)T \)
13 \( 1 + (0.550 + 0.834i)T \)
17 \( 1 + (-0.974 + 0.225i)T \)
19 \( 1 + (0.847 - 0.530i)T \)
23 \( 1 + (-0.851 - 0.524i)T \)
29 \( 1 + (-0.295 + 0.955i)T \)
31 \( 1 + (-0.909 + 0.415i)T \)
37 \( 1 + (-0.931 + 0.364i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.650 - 0.759i)T \)
47 \( 1 + (0.485 + 0.873i)T \)
53 \( 1 + (-0.141 + 0.989i)T \)
59 \( 1 + (0.943 - 0.331i)T \)
61 \( 1 + (0.106 + 0.994i)T \)
67 \( 1 + (-0.999 + 0.0345i)T \)
71 \( 1 + (-0.891 + 0.452i)T \)
73 \( 1 + (-0.696 - 0.718i)T \)
79 \( 1 + (-0.923 + 0.383i)T \)
83 \( 1 + (0.399 - 0.916i)T \)
89 \( 1 + (-0.978 + 0.205i)T \)
97 \( 1 + (-0.584 + 0.811i)T \)
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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

−17.68493669839442285996193085971, −17.11014334948903230922396649257, −15.999569444848193666181584571291, −15.34503558632608717969111347817, −14.67856113591176938202316012030, −13.81147635070043573319810297786, −13.32905903572616992803744209778, −12.72295377833107838091856773622, −11.73668778347608735782519540691, −11.564213552297665894152277823093, −10.68609743104215977407616457806, −9.93382578711369360871524029759, −9.26668712011019411166449129541, −8.62697101336160305441943320292, −8.025695179958600532520576680631, −7.393742461341768122248485852517, −6.698756515943714648150997718019, −5.88862869479691188998072052632, −5.37261256365640356930511967247, −3.85328220291700514697094014355, −3.20704341806711040457185251655, −2.4357982453465757904982142744, −1.90262131293031866367033027552, −1.37915499549415461379418162252, −0.05510169299574520050886752287, 1.38340728924773204801923313140, 1.763808724636782024280893620384, 2.77223952713942232087893561300, 3.95081907037789137511287925335, 4.61423002530976575992870561062, 5.176428146138103885546264479258, 5.90611515215908440792423150322, 6.78716050655130353646765311607, 7.430950106279754686526099576867, 8.44822532719130970768337728400, 8.76876979359732843164634637139, 9.28596107606332392837154375578, 10.12664463356372773464726670444, 10.53610442108136011300807954879, 11.20934691061980261019091666448, 11.901059241888636123511041037680, 13.25591394107478791420914875481, 13.77621043517780074892041715959, 14.238568011931373332506951594455, 14.87716397168311541867856956791, 15.7479378387092943389809846794, 16.28318315703473931512125962343, 16.673147209737566155229557216675, 17.39648547700419131726308189447, 17.88128610964958878160301423332

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