↑

Properties

Label 1-6027-6027.1280-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.907 - 0.420i$
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  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (0.623 + 0.781i)8-s + (−0.623 + 0.781i)10-s + (−0.974 + 0.222i)11-s + (−0.974 + 0.222i)13-s + (0.623 − 0.781i)16-s + (−0.433 + 0.900i)17-s + i·19-s + (0.900 + 0.433i)20-s + (0.433 + 0.900i)22-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (0.433 + 0.900i)26-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (0.623 + 0.781i)8-s + (−0.623 + 0.781i)10-s + (−0.974 + 0.222i)11-s + (−0.974 + 0.222i)13-s + (0.623 − 0.781i)16-s + (−0.433 + 0.900i)17-s + i·19-s + (0.900 + 0.433i)20-s + (0.433 + 0.900i)22-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (0.433 + 0.900i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.907 - 0.420i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1280, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.907 - 0.420i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04380113578 - 0.1988870438i\)
\(L(\frac12)\) \(\approx\) \(0.04380113578 - 0.1988870438i\)
\(L(1)\) \(\approx\) \(0.5185228971 - 0.2171504565i\)
\(L(1)\) \(\approx\) \(0.5185228971 - 0.2171504565i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (-0.974 + 0.222i)T \)
13 \( 1 + (-0.974 + 0.222i)T \)
17 \( 1 + (-0.433 + 0.900i)T \)
19 \( 1 + iT \)
23 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (-0.433 + 0.900i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.974 + 0.222i)T \)
53 \( 1 + (0.433 + 0.900i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.433 + 0.900i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.974 - 0.222i)T \)
97 \( 1 + iT \)
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

−18.09191436556047983205674281753, −17.316294194930854408972141126816, −16.63385128211357002150745869193, −15.87569120094462772919148211244, −15.361032097131062712719268062767, −14.969207441425764533828015639761, −14.27141996487111536075199632640, −13.34929483427759080160838956356, −13.15804482954689249751089075823, −11.980323573602142651005222236832, −11.33686602269269505708061376170, −10.56422550948317533155117082832, −10.01292122078464571298978950915, −9.20530423306822089912784746191, −8.54570218660884606303181560428, −7.70286249159742576430865473536, −7.217443084738802792130701969131, −6.861902498022778953569326588201, −5.84335692260215208307960153503, −5.056348914147028257539999760722, −4.66877210357655772570497675587, −3.569184019417275170121330753453, −2.899485889740140890343805726735, −2.03517056884974350120380078216, −0.51704255574831223970733447405, 0.103649396698279610340775012352, 1.36002685170598866063154034074, 1.94005044340938649529519534981, 2.86793402365969865578982906886, 3.6497975724509909167162995833, 4.30931575371976229192004754097, 5.08311128124940836718387486956, 5.45092556221841695561349176061, 6.84017025329686856096783340871, 7.63655473278594742016856958310, 8.17222684061166440240479125440, 8.81225687081523741119887923092, 9.48459707909824622450092870241, 10.1906516639777425928580821830, 10.88090991867960518676988739612, 11.40118930904206617565534474481, 12.40238999797211794890585058809, 12.63363079602812978523528349777, 13.05640441925800646358214877997, 14.08712557876361349575684600225, 14.76493400128198289293557771499, 15.42588839204636013230831191406, 16.39889763365801013380780213, 16.80483007430346891186355640847, 17.43691003629879405988827532812

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