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Properties

Label 1-6017-6017.2016-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.746 - 0.665i$
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  + (−0.906 − 0.421i)2-s + (−0.809 + 0.587i)3-s + (0.644 + 0.764i)4-s + (0.779 − 0.626i)5-s + (0.981 − 0.192i)6-s + (0.485 − 0.873i)7-s + (−0.262 − 0.964i)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 + (−0.262 + 0.964i)15-s + (−0.168 + 0.985i)16-s + (−0.607 + 0.794i)17-s + (−0.681 + 0.732i)18-s + ⋯
L(s)  = 1  + (−0.906 − 0.421i)2-s + (−0.809 + 0.587i)3-s + (0.644 + 0.764i)4-s + (0.779 − 0.626i)5-s + (0.981 − 0.192i)6-s + (0.485 − 0.873i)7-s + (−0.262 − 0.964i)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 + (−0.262 + 0.964i)15-s + (−0.168 + 0.985i)16-s + (−0.607 + 0.794i)17-s + (−0.681 + 0.732i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (2016, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.746 - 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9039698010 - 0.3447046405i\)
\(L(\frac12)\) \(\approx\) \(0.9039698010 - 0.3447046405i\)
\(L(1)\) \(\approx\) \(0.6582126630 - 0.1292801945i\)
\(L(1)\) \(\approx\) \(0.6582126630 - 0.1292801945i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.906 - 0.421i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.779 - 0.626i)T \)
7 \( 1 + (0.485 - 0.873i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.607 + 0.794i)T \)
19 \( 1 + (-0.443 + 0.896i)T \)
23 \( 1 + (-0.748 + 0.663i)T \)
29 \( 1 + (0.215 + 0.976i)T \)
31 \( 1 + (0.715 - 0.698i)T \)
37 \( 1 + (0.215 + 0.976i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (0.995 + 0.0965i)T \)
53 \( 1 + (-0.998 + 0.0483i)T \)
59 \( 1 + (-0.443 - 0.896i)T \)
61 \( 1 + (0.981 - 0.192i)T \)
67 \( 1 + (-0.970 - 0.239i)T \)
71 \( 1 + (0.715 + 0.698i)T \)
73 \( 1 + (0.836 - 0.548i)T \)
79 \( 1 + (-0.527 + 0.849i)T \)
83 \( 1 + (0.779 - 0.626i)T \)
89 \( 1 + (0.120 + 0.992i)T \)
97 \( 1 + (0.958 - 0.285i)T \)
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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

−17.97903804176434720803707432206, −17.29677910302997439288383998426, −16.71003393103525798202684114446, −15.85802926668665260836096915949, −15.45149429667347830391701224125, −14.51354993562094123551116193275, −13.92123952476887175543089453049, −13.36479094298266878704199725675, −12.23526525562401582257452719703, −11.66149580195753962811858588444, −11.15115632880678635172176804432, −10.537126527319231937607632671840, −9.81574360293103936804931929245, −9.01768818255191318488783722492, −8.53324658423544703780446149099, −7.601280125811131345061768788437, −6.88085362270197873842498191179, −6.40374471602360631773003990735, −5.92333708689193942939707970266, −5.10568111287694000767125701630, −4.49293164484977753476133854773, −2.78454062866588245814307400656, −2.141666049324477556596122350473, −1.78520247063060219087198420227, −0.630438909566871163342440133299, 0.57935239358582225815436946148, 1.376253724162471656834615896, 1.90666129965922811016360165520, 3.21296367508001025023213723502, 3.89995951937913498008164802384, 4.61843536817836864032042802428, 5.428512151175718663094985712833, 6.27144157229140036928878734824, 6.681435349941205294971302561508, 7.94414943213804448785989653928, 8.23927890692382638145706605412, 9.12972020088080569207419671255, 9.94380183706286568343199702369, 10.27993201807490326652032393917, 10.773324355603466780545966666420, 11.52956737007897225035299241630, 12.214825855270407237551237658035, 12.92154681361391272744612926287, 13.426104368441564849034232206675, 14.43701238162632172246164712555, 15.33871990047986970451892517179, 15.85836906786321816524710877934, 16.75498502817201502557112702492, 17.01565070811182307613974212764, 17.514838924364597315473817686750

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