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Properties

Label 1-6017-6017.3910-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.770 + 0.637i$
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  + (−0.963 − 0.266i)2-s + (−0.995 + 0.0896i)3-s + (0.858 + 0.512i)4-s + (0.473 + 0.880i)5-s + (0.983 + 0.178i)6-s + (−0.995 − 0.0896i)7-s + (−0.691 − 0.722i)8-s + (0.983 − 0.178i)9-s + (−0.222 − 0.974i)10-s + (−0.900 − 0.433i)12-s + (−0.0448 + 0.998i)13-s + (0.936 + 0.351i)14-s + (−0.550 − 0.834i)15-s + (0.473 + 0.880i)16-s + (0.473 + 0.880i)17-s + (−0.995 − 0.0896i)18-s + ⋯
L(s)  = 1  + (−0.963 − 0.266i)2-s + (−0.995 + 0.0896i)3-s + (0.858 + 0.512i)4-s + (0.473 + 0.880i)5-s + (0.983 + 0.178i)6-s + (−0.995 − 0.0896i)7-s + (−0.691 − 0.722i)8-s + (0.983 − 0.178i)9-s + (−0.222 − 0.974i)10-s + (−0.900 − 0.433i)12-s + (−0.0448 + 0.998i)13-s + (0.936 + 0.351i)14-s + (−0.550 − 0.834i)15-s + (0.473 + 0.880i)16-s + (0.473 + 0.880i)17-s + (−0.995 − 0.0896i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.770 + 0.637i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (3910, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2223601849 + 0.6177449506i\)
\(L(\frac12)\) \(\approx\) \(0.2223601849 + 0.6177449506i\)
\(L(1)\) \(\approx\) \(0.5103666625 + 0.1661513476i\)
\(L(1)\) \(\approx\) \(0.5103666625 + 0.1661513476i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.963 - 0.266i)T \)
3 \( 1 + (-0.995 + 0.0896i)T \)
5 \( 1 + (0.473 + 0.880i)T \)
7 \( 1 + (-0.995 - 0.0896i)T \)
13 \( 1 + (-0.0448 + 0.998i)T \)
17 \( 1 + (0.473 + 0.880i)T \)
19 \( 1 + (0.936 + 0.351i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (0.134 + 0.990i)T \)
31 \( 1 + (-0.393 - 0.919i)T \)
37 \( 1 + (-0.691 + 0.722i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.0448 + 0.998i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.473 + 0.880i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (0.473 + 0.880i)T \)
73 \( 1 + (0.936 - 0.351i)T \)
79 \( 1 + (0.753 + 0.657i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (-0.0448 + 0.998i)T \)
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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

−17.45466549044764117622475057372, −16.83419912521582776341275380141, −16.28812172412363905673592132869, −15.74924033773325576038029645446, −15.390703123671388457461433197499, −14.11723681754029150914695615444, −13.35758128339369327278384458848, −12.72419002648432480485990534433, −12.067968745317846288178483138261, −11.54392356667600844885089734691, −10.625051969194026418781453004682, −10.02249627220156818783235164563, −9.53092295923229246203906187139, −9.003274025325282250059727711657, −8.02616585573184075238014027050, −7.27288567710635639507972556309, −6.79041086411336127717294994359, −5.816127496303263300296326971322, −5.476662757182576467876804980, −4.94421722775697186691246221310, −3.59788091590945307063421812228, −2.75103134611065468047548040830, −1.75480775792965700889454530390, −0.84475649901316431441873286905, −0.39181368713012103541065321108, 0.985194779770486676529633557176, 1.703486125491820394858800367030, 2.64572095545012157972098969211, 3.42653945140773097138000161674, 4.060502493837533119793680047872, 5.31351731241410957654271151679, 6.11491481735747757734750613438, 6.55777114522063984878359947259, 7.08486065865617269704669961002, 7.74777523878176369705349954708, 8.930930851586261467193856312878, 9.52596026035180234120223758835, 10.08812621863962735143988264516, 10.57905128580195491611720950548, 11.183654396232290903470802841494, 11.87786254766520131437346619894, 12.51390914262967162772236636549, 13.11287599360484188189845786299, 14.0525394735880223570792135232, 14.89535468056167454384542708638, 15.61896013377240730309650526637, 16.25324625085411994436606074091, 16.96403702526746859585743112781, 17.06060894959653333614928147768, 18.15612031523765298967155728305

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