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Properties

Label 1-6017-6017.1994-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.258 - 0.966i$
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  + (0.981 + 0.192i)2-s + (−0.809 + 0.587i)3-s + (0.926 + 0.377i)4-s + (−0.998 − 0.0483i)5-s + (−0.906 + 0.421i)6-s + (−0.681 + 0.732i)7-s + (0.836 + 0.548i)8-s + (0.309 − 0.951i)9-s + (−0.970 − 0.239i)10-s + (−0.970 + 0.239i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.836 − 0.548i)15-s + (0.715 + 0.698i)16-s + (0.958 + 0.285i)17-s + (0.485 − 0.873i)18-s + ⋯
L(s)  = 1  + (0.981 + 0.192i)2-s + (−0.809 + 0.587i)3-s + (0.926 + 0.377i)4-s + (−0.998 − 0.0483i)5-s + (−0.906 + 0.421i)6-s + (−0.681 + 0.732i)7-s + (0.836 + 0.548i)8-s + (0.309 − 0.951i)9-s + (−0.970 − 0.239i)10-s + (−0.970 + 0.239i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.836 − 0.548i)15-s + (0.715 + 0.698i)16-s + (0.958 + 0.285i)17-s + (0.485 − 0.873i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.258 - 0.966i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1994, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.258 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1411711291 - 0.1839055901i\)
\(L(\frac12)\) \(\approx\) \(0.1411711291 - 0.1839055901i\)
\(L(1)\) \(\approx\) \(0.9851477161 + 0.3308001257i\)
\(L(1)\) \(\approx\) \(0.9851477161 + 0.3308001257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.981 + 0.192i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.998 - 0.0483i)T \)
7 \( 1 + (-0.681 + 0.732i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.958 + 0.285i)T \)
19 \( 1 + (-0.989 + 0.144i)T \)
23 \( 1 + (-0.748 - 0.663i)T \)
29 \( 1 + (0.995 - 0.0965i)T \)
31 \( 1 + (-0.168 - 0.985i)T \)
37 \( 1 + (0.995 - 0.0965i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.970 + 0.239i)T \)
47 \( 1 + (0.215 - 0.976i)T \)
53 \( 1 + (0.779 + 0.626i)T \)
59 \( 1 + (-0.989 - 0.144i)T \)
61 \( 1 + (-0.906 + 0.421i)T \)
67 \( 1 + (-0.970 + 0.239i)T \)
71 \( 1 + (-0.168 + 0.985i)T \)
73 \( 1 + (-0.262 + 0.964i)T \)
79 \( 1 + (-0.0724 + 0.997i)T \)
83 \( 1 + (-0.998 - 0.0483i)T \)
89 \( 1 + (0.120 - 0.992i)T \)
97 \( 1 + (-0.607 - 0.794i)T \)
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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

−17.955614605700175320366429992766, −16.90344728384464776326525126309, −16.453781105957736950647837708011, −16.07777901466285986809482112372, −15.33248806375665050452859779639, −14.48367219252528390696430672438, −13.743720212226766573567144310896, −13.36710468872252738018474769174, −12.39376769103982933673439699333, −12.13489658631078601654321240041, −11.5584149813684354966649740838, −10.69094023422084835487457085265, −10.45899360100433248677248395887, −9.433433537082340248213592916291, −8.20898965503491291447495213806, −7.56571354658412438741706665464, −6.89324484215564507206449980831, −6.4926968928275650538315331647, −5.77566839444211178580015409212, −4.764009182807694528309583236890, −4.34656286534895241139333288930, −3.557524132116575228330503984024, −2.90034250888694342097868581850, −1.74791955892267477333682066716, −1.071874404499823437877001197619, 0.049889390908332564988658200991, 1.269791121635661670720051374, 2.65121480265848743988522231937, 3.19054974010289246863451253433, 4.001961489986038157288877924594, 4.41568026192127858773102186547, 5.349607722293017557730265318307, 5.91867401895502999167552795020, 6.40960687374078039417695249995, 7.2172164315248191904695160180, 8.17983516683751096004820038908, 8.5577777616082483386458078422, 9.87521756078154733868956487526, 10.356678679200327523019718338263, 11.07743132687899235367281330640, 11.82064453450891549102812174340, 12.23175773937643280868981972261, 12.72719245687626566032261395710, 13.38301516727092869357978907574, 14.61770948208410386000946081681, 15.051122597050279370477547968576, 15.46615383040716038634879334368, 16.11852507497932724884782779855, 16.63696846958344210016043386424, 17.100898611144216759916906316128

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