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Properties

Label 1-6017-6017.798-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.625 + 0.780i$
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  + (−0.293 + 0.955i)2-s + (−0.309 − 0.951i)3-s + (−0.827 − 0.561i)4-s + (−0.0563 − 0.998i)5-s + (0.999 − 0.0161i)6-s + (0.0884 + 0.996i)7-s + (0.779 − 0.626i)8-s + (−0.809 + 0.587i)9-s + (0.970 + 0.239i)10-s + (−0.278 + 0.960i)12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (−0.932 + 0.362i)15-s + (0.369 + 0.929i)16-s + (0.759 + 0.650i)17-s + (−0.324 − 0.945i)18-s + ⋯
L(s)  = 1  + (−0.293 + 0.955i)2-s + (−0.309 − 0.951i)3-s + (−0.827 − 0.561i)4-s + (−0.0563 − 0.998i)5-s + (0.999 − 0.0161i)6-s + (0.0884 + 0.996i)7-s + (0.779 − 0.626i)8-s + (−0.809 + 0.587i)9-s + (0.970 + 0.239i)10-s + (−0.278 + 0.960i)12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (−0.932 + 0.362i)15-s + (0.369 + 0.929i)16-s + (0.759 + 0.650i)17-s + (−0.324 − 0.945i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.625 + 0.780i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (798, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.625 + 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1735011714 + 0.3615017003i\)
\(L(\frac12)\) \(\approx\) \(0.1735011714 + 0.3615017003i\)
\(L(1)\) \(\approx\) \(0.6484154598 + 0.06907318621i\)
\(L(1)\) \(\approx\) \(0.6484154598 + 0.06907318621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.293 + 0.955i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.0563 - 0.998i)T \)
7 \( 1 + (0.0884 + 0.996i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (0.759 + 0.650i)T \)
19 \( 1 + (0.769 + 0.638i)T \)
23 \( 1 + (-0.948 + 0.316i)T \)
29 \( 1 + (-0.399 - 0.916i)T \)
31 \( 1 + (0.443 - 0.896i)T \)
37 \( 1 + (-0.594 + 0.804i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (0.692 + 0.721i)T \)
47 \( 1 + (0.870 - 0.493i)T \)
53 \( 1 + (-0.704 + 0.709i)T \)
59 \( 1 + (0.769 - 0.638i)T \)
61 \( 1 + (-0.999 + 0.0161i)T \)
67 \( 1 + (0.692 + 0.721i)T \)
71 \( 1 + (0.554 - 0.832i)T \)
73 \( 1 + (-0.541 - 0.840i)T \)
79 \( 1 + (0.644 - 0.764i)T \)
83 \( 1 + (-0.892 - 0.450i)T \)
89 \( 1 + (-0.120 + 0.992i)T \)
97 \( 1 + (-0.877 - 0.478i)T \)
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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

−17.66233307727702513597772219599, −16.91250404775911231250437966677, −16.28880760718882958587830279410, −15.66860702667414883296138695394, −14.55225947935381851150052613910, −14.13457626441478348157710390677, −13.86620782180910198481072499737, −12.56075267803703468296931634267, −12.02766630982484084707103649311, −11.22473483534496284460276917156, −10.9272941937027606918273843756, −10.129519116978694034401308405, −9.820783474356646404692858204624, −9.18786042450069565238452068651, −8.20753696892555060167854244244, −7.32940298432966709967982774595, −6.97015051672881472455053618685, −5.72295096189226554311125500773, −4.97378848890963758373429904230, −4.28977943251416898466910085561, −3.644378240441440078531099232484, −3.01536740571979831570304613512, −2.37093167010821112373261410456, −1.17483181615757459560174084215, −0.15358698071657867161827611209, 0.87241662788690263796478434435, 1.6600720640967479685410009422, 2.41223805217816546826299513741, 3.66787946937607499503193206589, 4.65934867894831709297958219805, 5.38677639350559423778956146897, 5.80529225389108007810097873529, 6.26416356438103935988812416372, 7.485332924283552531886374168619, 7.88371366799180926695298327366, 8.25195560336848345905426804694, 9.19383336356926225927034616510, 9.67744173101792800119436786217, 10.498366088966576392333371425868, 11.73756477119863695662485172659, 12.11923871571907024675227972942, 12.6846099034034734367624652481, 13.35712030153285479954838634494, 14.06564314185092954502383691355, 14.68661472932289885547484417719, 15.50512895254433940655080282878, 16.000090013574145990154731953892, 16.85922287116150597702756295565, 17.21658796138703462479364990877, 17.77563416177256530647811060816

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