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Properties

Label 1-6017-6017.1196-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.372 + 0.927i$
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  + (−0.657 + 0.753i)2-s + (0.809 − 0.587i)3-s + (−0.136 − 0.990i)4-s + (−0.457 − 0.889i)5-s + (−0.0884 + 0.996i)6-s + (−0.293 − 0.955i)7-s + (0.836 + 0.548i)8-s + (0.309 − 0.951i)9-s + (0.970 + 0.239i)10-s + (−0.692 − 0.721i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (−0.892 − 0.450i)15-s + (−0.962 + 0.270i)16-s + (−0.726 + 0.686i)17-s + (0.513 + 0.857i)18-s + ⋯
L(s)  = 1  + (−0.657 + 0.753i)2-s + (0.809 − 0.587i)3-s + (−0.136 − 0.990i)4-s + (−0.457 − 0.889i)5-s + (−0.0884 + 0.996i)6-s + (−0.293 − 0.955i)7-s + (0.836 + 0.548i)8-s + (0.309 − 0.951i)9-s + (0.970 + 0.239i)10-s + (−0.692 − 0.721i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (−0.892 − 0.450i)15-s + (−0.962 + 0.270i)16-s + (−0.726 + 0.686i)17-s + (0.513 + 0.857i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.372 + 0.927i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.372 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2062596657 - 0.3051085249i\)
\(L(\frac12)\) \(\approx\) \(-0.2062596657 - 0.3051085249i\)
\(L(1)\) \(\approx\) \(0.6457011702 - 0.2749300914i\)
\(L(1)\) \(\approx\) \(0.6457011702 - 0.2749300914i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.657 + 0.753i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.457 - 0.889i)T \)
7 \( 1 + (-0.293 - 0.955i)T \)
13 \( 1 + (-0.669 - 0.743i)T \)
17 \( 1 + (-0.726 + 0.686i)T \)
19 \( 1 + (-0.619 - 0.784i)T \)
23 \( 1 + (0.200 - 0.979i)T \)
29 \( 1 + (-0.995 + 0.0965i)T \)
31 \( 1 + (0.168 + 0.985i)T \)
37 \( 1 + (0.414 - 0.910i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.278 - 0.960i)T \)
47 \( 1 + (-0.953 + 0.301i)T \)
53 \( 1 + (-0.932 + 0.362i)T \)
59 \( 1 + (-0.619 + 0.784i)T \)
61 \( 1 + (0.0884 - 0.996i)T \)
67 \( 1 + (0.278 - 0.960i)T \)
71 \( 1 + (0.769 + 0.638i)T \)
73 \( 1 + (0.704 + 0.709i)T \)
79 \( 1 + (-0.0724 + 0.997i)T \)
83 \( 1 + (0.541 - 0.840i)T \)
89 \( 1 + (-0.120 + 0.992i)T \)
97 \( 1 + (-0.384 + 0.923i)T \)
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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

−18.50077480974519710201202008756, −17.72602741415044807935245746275, −16.74997334666534204892959553675, −16.25577781013262418128112000655, −15.47357507488058216037241159877, −14.97443525849346929233404470387, −14.36300450875560735113643984530, −13.45867379762635418759873634751, −12.92975036785455726902865141750, −11.96670331739610100875318821731, −11.39282027251794964159511947341, −11.00148608006564046631291042120, −9.86986568771900965823262342568, −9.67219977596967621222392148463, −9.06277523080073850825679678921, −8.163708595281911048707275989719, −7.7708322074163532525201283187, −6.9452061851580231523313840999, −6.17625624697205387869491412039, −4.90270655480868842850115570067, −4.27490577102927549286872089147, −3.462853025767079680016362969950, −2.93241170190291370473767145928, −2.2474464182165625128900237083, −1.73833212235805231480858991573, 0.130142158782712477821563221108, 0.744564478467084517663133738940, 1.662331418083663597137727157779, 2.44612647836246495120086132729, 3.594416674984564248014485475584, 4.31332568908342465952245893672, 4.96832147655695814591223944262, 5.93733338209466970679697247509, 6.89281631048565806083310707356, 7.10565792865504042944028036780, 8.10700842230875911203143213168, 8.26658501675635050730105740253, 9.20941595109909392266942543019, 9.52498680723585581689805666126, 10.63977058419949565379400452135, 10.958845044805965405637185017682, 12.32663262599188716041726434520, 12.78840785729487752237411430299, 13.34638063358492589080069620206, 14.07587434094711884983314221307, 14.746175334656219345099658281279, 15.39171810322299470078451777207, 15.86928144768149211831177924926, 16.76352402038106374003250624477, 17.273611093345665416950605079369

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