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Properties

Label 1-6027-6027.2642-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.892 - 0.451i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (0.842 + 0.538i)2-s + (0.420 + 0.907i)4-s + (−0.280 − 0.959i)5-s + (−0.134 + 0.990i)8-s + (0.280 − 0.959i)10-s + (0.163 + 0.986i)11-s + (0.0448 − 0.998i)13-s + (−0.646 + 0.762i)16-s + (0.575 + 0.817i)17-s + (−0.913 + 0.406i)19-s + (0.753 − 0.657i)20-s + (−0.393 + 0.919i)22-s + (0.946 − 0.323i)23-s + (−0.842 + 0.538i)25-s + (0.575 − 0.817i)26-s + ⋯
L(s)  = 1  + (0.842 + 0.538i)2-s + (0.420 + 0.907i)4-s + (−0.280 − 0.959i)5-s + (−0.134 + 0.990i)8-s + (0.280 − 0.959i)10-s + (0.163 + 0.986i)11-s + (0.0448 − 0.998i)13-s + (−0.646 + 0.762i)16-s + (0.575 + 0.817i)17-s + (−0.913 + 0.406i)19-s + (0.753 − 0.657i)20-s + (−0.393 + 0.919i)22-s + (0.946 − 0.323i)23-s + (−0.842 + 0.538i)25-s + (0.575 − 0.817i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.892 - 0.451i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (2642, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.892 - 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1108619630 + 0.4647705157i\)
\(L(\frac12)\) \(\approx\) \(-0.1108619630 + 0.4647705157i\)
\(L(1)\) \(\approx\) \(1.230952190 + 0.4179545697i\)
\(L(1)\) \(\approx\) \(1.230952190 + 0.4179545697i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.842 + 0.538i)T \)
5 \( 1 + (-0.280 - 0.959i)T \)
11 \( 1 + (0.163 + 0.986i)T \)
13 \( 1 + (0.0448 - 0.998i)T \)
17 \( 1 + (0.575 + 0.817i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.946 - 0.323i)T \)
29 \( 1 + (-0.858 - 0.512i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.873 + 0.486i)T \)
43 \( 1 + (0.473 - 0.880i)T \)
47 \( 1 + (-0.842 - 0.538i)T \)
53 \( 1 + (-0.420 - 0.907i)T \)
59 \( 1 + (-0.999 + 0.0299i)T \)
61 \( 1 + (-0.193 - 0.981i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (-0.858 + 0.512i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.887 + 0.460i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−17.321220363200220654415428703660, −16.3780175616278167019090096978, −16.04530960001574825051199078741, −14.992192073135621333581652505048, −14.72632569134325619727329358202, −14.00822592929939898614206152074, −13.501912341866170220074400829264, −12.78994008162061970479546675742, −11.875761357555425263434149264044, −11.41673916710807445995993698413, −10.89683344152189296253527040495, −10.38886164001850972504444096916, −9.262889582547318225938343419050, −9.024486633719528617678439707019, −7.58579893047526748343422795734, −7.19798871542612317278895501399, −6.26930360595894597464504416404, −5.94077793957135402943393748515, −4.900132946167726642236061305966, −4.25092784640360808047736017745, −3.36634415392424373797066917840, −3.04508798287451751407403862662, −2.12666407616192513669940696861, −1.32852915823601056868792896419, −0.07895605843345936322217280789, 1.417981190627866254706894877332, 2.07331124008608559869292796391, 3.211392316993123710861202661962, 3.82755287233750897522546235099, 4.51452078294656005685686003878, 5.21776526057299324422121874891, 5.68377760913567614101732703905, 6.59874904320702992743075838164, 7.30551540626176165307137966409, 8.02831372620706624848058680792, 8.49131313466553139379804746963, 9.25690154088660888421703758807, 10.22363123242516103843430257167, 10.89655557989656887410336466356, 11.81006222712436269508995978074, 12.52127636940282133870417274841, 12.73026494347353052816508199999, 13.31127772954569236439771342973, 14.269636741916408281306534952678, 15.10808618740895632687173465132, 15.1495850017678129163928348851, 16.08064905166322651396750310228, 16.7489861826974669453190481536, 17.26662192354470285239982219386, 17.66292121637234107082509180360

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