↑

Properties

Label 1-6027-6027.1622-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.999 + 0.0170i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.599 − 0.800i)2-s + (−0.280 − 0.959i)4-s + (−0.447 + 0.894i)5-s + (−0.936 − 0.351i)8-s + (0.447 + 0.894i)10-s + (0.420 − 0.907i)11-s + (−0.393 − 0.919i)13-s + (−0.842 + 0.538i)16-s + (−0.971 + 0.237i)17-s + (−0.104 + 0.994i)19-s + (0.983 + 0.178i)20-s + (−0.473 − 0.880i)22-s + (0.646 + 0.762i)23-s + (−0.599 − 0.800i)25-s + (−0.971 − 0.237i)26-s + ⋯
L(s)  = 1  + (0.599 − 0.800i)2-s + (−0.280 − 0.959i)4-s + (−0.447 + 0.894i)5-s + (−0.936 − 0.351i)8-s + (0.447 + 0.894i)10-s + (0.420 − 0.907i)11-s + (−0.393 − 0.919i)13-s + (−0.842 + 0.538i)16-s + (−0.971 + 0.237i)17-s + (−0.104 + 0.994i)19-s + (0.983 + 0.178i)20-s + (−0.473 − 0.880i)22-s + (0.646 + 0.762i)23-s + (−0.599 − 0.800i)25-s + (−0.971 − 0.237i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.999 + 0.0170i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.999 + 0.0170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008506372020 - 0.9963588577i\)
\(L(\frac12)\) \(\approx\) \(0.008506372020 - 0.9963588577i\)
\(L(1)\) \(\approx\) \(0.9623642253 - 0.5120332561i\)
\(L(1)\) \(\approx\) \(0.9623642253 - 0.5120332561i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.599 - 0.800i)T \)
5 \( 1 + (-0.447 + 0.894i)T \)
11 \( 1 + (0.420 - 0.907i)T \)
13 \( 1 + (-0.393 - 0.919i)T \)
17 \( 1 + (-0.971 + 0.237i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.646 + 0.762i)T \)
29 \( 1 + (0.134 + 0.990i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.791 - 0.611i)T \)
43 \( 1 + (-0.963 - 0.266i)T \)
47 \( 1 + (0.599 - 0.800i)T \)
53 \( 1 + (-0.280 - 0.959i)T \)
59 \( 1 + (0.712 - 0.701i)T \)
61 \( 1 + (0.337 + 0.941i)T \)
67 \( 1 + (-0.669 - 0.743i)T \)
71 \( 1 + (0.134 - 0.990i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.992 + 0.119i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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

−17.56606180578430485418172242603, −17.33066574667204272099788595965, −16.672400887609509344728615357609, −16.018660635345479587973824245486, −15.34594551208163990081759062148, −14.97823170840227013334609199498, −14.13736697351824940307073919128, −13.33495803290525716878151026043, −13.01884014476244797381672802830, −12.10176617031954461414008265778, −11.77442529401776975412283980840, −11.0592482585603706546152567209, −9.7073394491038240583685233347, −9.26411640821177942342321431187, −8.58490776337808522420419566956, −7.983020417535254427286811754284, −7.014777201002139619551233231667, −6.78432352914924049710420959798, −5.869191793666156803114491441071, −4.8529290849232116857977573416, −4.40783337408948952607021357432, −4.21970513924428768782490377584, −2.864421168502831756234730540189, −2.23762588714094790364912015626, −1.01669184493477670828332760158, 0.22370117377820550182992222920, 1.27098143041717089218122021686, 2.207048272044569537888805400627, 3.021133568261296781775752375588, 3.473795352557966911451399904555, 4.13835731969178603795860442059, 5.06467618240647211908005851646, 5.78398910637983964861394422998, 6.453362535483901560342744572146, 7.11058486008735610547885947681, 8.10254682223713430244169897364, 8.731964474014305402600952878956, 9.625042408792858082126307927093, 10.40910504218209793897271141457, 10.755073649871818931837279991008, 11.51004719317867633013457225746, 11.94446524110211818441312411491, 12.80215155584966646416137559509, 13.40230039343110800822166780518, 14.08642895269119023273572744711, 14.713959991135455944204437429636, 15.189581922133035672386116008, 15.82850571901971388764916587178, 16.67595423481552200466986043835, 17.673316904525188602528580813642

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