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Properties

Label 1-6017-6017.503-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.0784 + 0.996i$
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  + (0.909 − 0.415i)2-s + (−0.983 + 0.178i)3-s + (0.655 − 0.755i)4-s + (−0.335 + 0.942i)5-s + (−0.821 + 0.570i)6-s + (0.411 + 0.911i)7-s + (0.282 − 0.959i)8-s + (0.936 − 0.351i)9-s + (0.0862 + 0.996i)10-s + (−0.509 + 0.860i)12-s + (0.995 + 0.0896i)13-s + (0.753 + 0.657i)14-s + (0.161 − 0.986i)15-s + (−0.141 − 0.989i)16-s + (0.762 + 0.647i)17-s + (0.705 − 0.708i)18-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)2-s + (−0.983 + 0.178i)3-s + (0.655 − 0.755i)4-s + (−0.335 + 0.942i)5-s + (−0.821 + 0.570i)6-s + (0.411 + 0.911i)7-s + (0.282 − 0.959i)8-s + (0.936 − 0.351i)9-s + (0.0862 + 0.996i)10-s + (−0.509 + 0.860i)12-s + (0.995 + 0.0896i)13-s + (0.753 + 0.657i)14-s + (0.161 − 0.986i)15-s + (−0.141 − 0.989i)16-s + (0.762 + 0.647i)17-s + (0.705 − 0.708i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0784 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0784 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.0784 + 0.996i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.0784 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.849012596 + 1.709237365i\)
\(L(\frac12)\) \(\approx\) \(1.849012596 + 1.709237365i\)
\(L(1)\) \(\approx\) \(1.454544198 + 0.2675180904i\)
\(L(1)\) \(\approx\) \(1.454544198 + 0.2675180904i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.909 - 0.415i)T \)
3 \( 1 + (-0.983 + 0.178i)T \)
5 \( 1 + (-0.335 + 0.942i)T \)
7 \( 1 + (0.411 + 0.911i)T \)
13 \( 1 + (0.995 + 0.0896i)T \)
17 \( 1 + (0.762 + 0.647i)T \)
19 \( 1 + (0.127 + 0.991i)T \)
23 \( 1 + (0.650 + 0.759i)T \)
29 \( 1 + (0.977 + 0.212i)T \)
31 \( 1 + (-0.0379 + 0.999i)T \)
37 \( 1 + (-0.424 + 0.905i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.725 + 0.688i)T \)
47 \( 1 + (-0.989 + 0.144i)T \)
53 \( 1 + (0.988 - 0.151i)T \)
59 \( 1 + (-0.995 + 0.0965i)T \)
61 \( 1 + (0.373 + 0.927i)T \)
67 \( 1 + (0.509 - 0.860i)T \)
71 \( 1 + (-0.965 + 0.259i)T \)
73 \( 1 + (0.888 - 0.458i)T \)
79 \( 1 + (0.878 - 0.476i)T \)
83 \( 1 + (-0.527 - 0.849i)T \)
89 \( 1 + (-0.997 - 0.0689i)T \)
97 \( 1 + (-0.209 + 0.977i)T \)
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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

−17.312999091239431263848841315697, −16.76252352654210094788221688558, −16.29756267812581903556440153529, −15.70875823471399911791434793154, −15.12892423376695773392689586358, −13.96474060832621020583777376533, −13.645260478550753557607736331132, −12.92689108661068237225054283659, −12.401507221986829759412808328215, −11.69563105530125407991569904502, −11.15554491901962218170348192776, −10.63992060616938277627084058252, −9.64237708556196640055711229034, −8.57715947344219765045496853432, −8.025677043970118103373261954497, −7.17467038849073140248435524393, −6.83745537546158269413252604014, −5.838415666432535701341860562211, −5.26844283030303215821372286650, −4.66824955659834017325161735311, −4.13238795618326938793297010981, −3.41501591678763896233160065815, −2.20912951322147829693151236704, −1.1761250897964479240566354611, −0.57176065596621767003561871711, 1.29207970516334030281263685530, 1.634794702198249207798498847765, 2.94522922757945342025049399033, 3.4031214565184730709934081967, 4.17081422189849877404623653116, 5.030413915608926462199736886908, 5.61088321141411159125668597597, 6.26785617278065504026431359488, 6.67607999820462328869309854810, 7.63786072963186327748432051258, 8.4264137368444954997966176188, 9.57060323922670024450943763381, 10.26595160035859943542231786237, 10.7835654543389792764726998167, 11.33987901930079221148035186116, 12.042357865259285346522325109723, 12.23927282350519093674822863405, 13.21113847986956123105284463149, 13.93614268820038100819706849285, 14.80030094262040497581840966350, 15.04545936020821174166513828149, 15.90806142653181407681406083716, 16.196689936555851210783019244497, 17.22763997484815602757224801335, 18.13552176219794061428308735940

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