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Properties

Label 1-6017-6017.106-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.933 - 0.357i$
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  + (0.594 + 0.804i)2-s + (0.809 + 0.587i)3-s + (−0.293 + 0.955i)4-s + (0.726 + 0.686i)5-s + (0.00805 + 0.999i)6-s + (0.737 + 0.675i)7-s + (−0.943 + 0.331i)8-s + (0.309 + 0.951i)9-s + (−0.120 + 0.992i)10-s + (−0.799 + 0.600i)12-s + (0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (0.184 + 0.982i)15-s + (−0.827 − 0.561i)16-s + (0.937 + 0.347i)17-s + (−0.581 + 0.813i)18-s + ⋯
L(s)  = 1  + (0.594 + 0.804i)2-s + (0.809 + 0.587i)3-s + (−0.293 + 0.955i)4-s + (0.726 + 0.686i)5-s + (0.00805 + 0.999i)6-s + (0.737 + 0.675i)7-s + (−0.943 + 0.331i)8-s + (0.309 + 0.951i)9-s + (−0.120 + 0.992i)10-s + (−0.799 + 0.600i)12-s + (0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (0.184 + 0.982i)15-s + (−0.827 − 0.561i)16-s + (0.937 + 0.347i)17-s + (−0.581 + 0.813i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.933 - 0.357i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.933 - 0.357i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8478433968 + 4.582195275i\)
\(L(\frac12)\) \(\approx\) \(-0.8478433968 + 4.582195275i\)
\(L(1)\) \(\approx\) \(1.167786821 + 1.991213522i\)
\(L(1)\) \(\approx\) \(1.167786821 + 1.991213522i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.594 + 0.804i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.726 + 0.686i)T \)
7 \( 1 + (0.737 + 0.675i)T \)
13 \( 1 + (0.978 + 0.207i)T \)
17 \( 1 + (0.937 + 0.347i)T \)
19 \( 1 + (-0.339 + 0.940i)T \)
23 \( 1 + (-0.987 + 0.160i)T \)
29 \( 1 + (-0.836 - 0.548i)T \)
31 \( 1 + (0.527 + 0.849i)T \)
37 \( 1 + (0.892 - 0.450i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.919 - 0.391i)T \)
47 \( 1 + (0.966 - 0.254i)T \)
53 \( 1 + (-0.384 - 0.923i)T \)
59 \( 1 + (-0.339 - 0.940i)T \)
61 \( 1 + (-0.00805 - 0.999i)T \)
67 \( 1 + (-0.919 - 0.391i)T \)
71 \( 1 + (0.471 + 0.881i)T \)
73 \( 1 + (0.877 + 0.478i)T \)
79 \( 1 + (-0.906 + 0.421i)T \)
83 \( 1 + (-0.231 + 0.972i)T \)
89 \( 1 + (0.748 - 0.663i)T \)
97 \( 1 + (0.247 - 0.968i)T \)
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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

−17.58563116937881202417112142637, −16.77814206715598992416499813947, −15.920602263088865435146540370967, −15.01517188680314468687947471485, −14.49723950613154768913520164461, −13.81480075020336288093970387858, −13.39417115639901180831731686560, −13.01424999156941977681201692798, −12.149088841190917785409670427684, −11.58531758187303701018226031227, −10.72657828394461097427008719033, −10.08302578946892904250763209402, −9.34245585988105797013352957302, −8.82284574705578896574702843258, −8.02959422214846374500625941159, −7.37783033540232727651056941259, −6.200657232246659083819384681526, −5.91785705968019489148547799699, −4.81666339436535424591956712089, −4.29201213564122773117293373928, −3.52702050045964653169713560516, −2.66445189646472537479278760630, −2.00079041615048143591476551, −1.17593605798157061693270255668, −0.8517678256495445365956206270, 1.66892356040345719480803168238, 2.183526234105848022957033637793, 3.115380587699900281768013124925, 3.71888946858718879830248583342, 4.34660846704394841392141355551, 5.39895890026820908924910020331, 5.75358819370926591497986485786, 6.446700519313451399327264616248, 7.456015867385739413815631685155, 8.093736417856609216479110896437, 8.50485942587013077436167622439, 9.35989729181279904948807993470, 9.93532403718794216934455849599, 10.80215758786536785244711582055, 11.46198144638585472067256467765, 12.366042109919148828567868887526, 13.06139649373389535452242322828, 13.87297401510892899750580262043, 14.32685198386623165311589346479, 14.58049197595673573900585596611, 15.45314009932162254689769940787, 15.79859745651145745734772113902, 16.70485991691503478777120171308, 17.21187321175973721095709841646, 18.191200262495951494473553427278

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