↑

Properties

Label 1-6017-6017.1339-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.621 - 0.783i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.999 + 0.0322i)2-s + (0.809 − 0.587i)3-s + (0.997 + 0.0643i)4-s + (−0.870 + 0.493i)5-s + (0.827 − 0.561i)6-s + (−0.136 − 0.990i)7-s + (0.995 + 0.0965i)8-s + (0.309 − 0.951i)9-s + (−0.885 + 0.464i)10-s + (0.845 − 0.534i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (−0.414 + 0.910i)15-s + (0.991 + 0.128i)16-s + (0.457 + 0.889i)17-s + (0.339 − 0.940i)18-s + ⋯
L(s)  = 1  + (0.999 + 0.0322i)2-s + (0.809 − 0.587i)3-s + (0.997 + 0.0643i)4-s + (−0.870 + 0.493i)5-s + (0.827 − 0.561i)6-s + (−0.136 − 0.990i)7-s + (0.995 + 0.0965i)8-s + (0.309 − 0.951i)9-s + (−0.885 + 0.464i)10-s + (0.845 − 0.534i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (−0.414 + 0.910i)15-s + (0.991 + 0.128i)16-s + (0.457 + 0.889i)17-s + (0.339 − 0.940i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.621 - 0.783i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.621 - 0.783i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.478488052 - 2.162731489i\)
\(L(\frac12)\) \(\approx\) \(4.478488052 - 2.162731489i\)
\(L(1)\) \(\approx\) \(2.414289103 - 0.5776162007i\)
\(L(1)\) \(\approx\) \(2.414289103 - 0.5776162007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.999 + 0.0322i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.870 + 0.493i)T \)
7 \( 1 + (-0.136 - 0.990i)T \)
13 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (0.457 + 0.889i)T \)
19 \( 1 + (0.877 + 0.478i)T \)
23 \( 1 + (-0.799 - 0.600i)T \)
29 \( 1 + (-0.485 + 0.873i)T \)
31 \( 1 + (-0.958 + 0.285i)T \)
37 \( 1 + (0.999 - 0.0161i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.0402 - 0.999i)T \)
47 \( 1 + (0.974 - 0.223i)T \)
53 \( 1 + (0.594 - 0.804i)T \)
59 \( 1 + (0.877 - 0.478i)T \)
61 \( 1 + (-0.827 + 0.561i)T \)
67 \( 1 + (-0.0402 - 0.999i)T \)
71 \( 1 + (0.231 + 0.972i)T \)
73 \( 1 + (-0.737 + 0.675i)T \)
79 \( 1 + (0.715 - 0.698i)T \)
83 \( 1 + (-0.00805 + 0.999i)T \)
89 \( 1 + (0.970 - 0.239i)T \)
97 \( 1 + (-0.932 + 0.362i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.92914667437511907929286012188, −16.50006572562693474539347469222, −16.27713876136504242659671706925, −15.69435264252986498962081057687, −15.234370365518125715294647813838, −14.63808935191504018121939847723, −13.809714643067104512876916124930, −13.35880381838488583234766461563, −12.61655073030083404607604111927, −11.85763997093854638149525206576, −11.40423841730416938917651304625, −10.79035074577157260476930854393, −9.57204941076655088846028118874, −9.291475178390806655505021485048, −8.37059630185853537579945966225, −7.67908355467669692667061383025, −7.25094720187153736653836568764, −5.92418119949363344458500562541, −5.55105033012268000125542907393, −4.663761031183427262136123884350, −4.09228233946764212102574717387, −3.42336104591749949943047724383, −2.827623712811660166079220564904, −2.05952550828472863305885633046, −1.061993871438358806591561098095, 0.86239373004035843598221664183, 1.586419047460894254400973893764, 2.558857530272687211564275739177, 3.42120989164477887868316102040, 3.78037838056426037626269353447, 4.185744900600891785183971136438, 5.46767620596982098027355342438, 6.24181801505252854862211119858, 6.875783364872339435759919608953, 7.52072397287980905983924210948, 7.9277317622353883182483954908, 8.63008323208856585398241078826, 9.8036014723575697013925457901, 10.59710186173020973273111380166, 11.01954438146687439396364692570, 11.93827078929041235399503447503, 12.47631717779177676364460219248, 13.11967241915459287993108319368, 13.708167814397502247096747116, 14.4618010989768679411330245682, 14.64180699714480631696021915729, 15.49134259708427885381651284703, 16.22905233578475335975347771641, 16.56570694359080327347140322030, 17.75280992045046254840400813139

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