↑

Properties

Label 1-6027-6027.1952-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.341 - 0.939i$
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  + (−0.753 + 0.657i)2-s + (0.134 − 0.990i)4-s + (0.936 + 0.351i)5-s + (0.550 + 0.834i)8-s + (−0.936 + 0.351i)10-s + (0.858 + 0.512i)11-s + (0.753 − 0.657i)13-s + (−0.963 − 0.266i)16-s + (−0.134 − 0.990i)17-s + (−0.809 + 0.587i)19-s + (0.473 − 0.880i)20-s + (−0.983 + 0.178i)22-s + (−0.473 − 0.880i)23-s + (0.753 + 0.657i)25-s + (−0.134 + 0.990i)26-s + ⋯
L(s)  = 1  + (−0.753 + 0.657i)2-s + (0.134 − 0.990i)4-s + (0.936 + 0.351i)5-s + (0.550 + 0.834i)8-s + (−0.936 + 0.351i)10-s + (0.858 + 0.512i)11-s + (0.753 − 0.657i)13-s + (−0.963 − 0.266i)16-s + (−0.134 − 0.990i)17-s + (−0.809 + 0.587i)19-s + (0.473 − 0.880i)20-s + (−0.983 + 0.178i)22-s + (−0.473 − 0.880i)23-s + (0.753 + 0.657i)25-s + (−0.134 + 0.990i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.341 - 0.939i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1952, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.341 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7791638880 - 0.5460032604i\)
\(L(\frac12)\) \(\approx\) \(0.7791638880 - 0.5460032604i\)
\(L(1)\) \(\approx\) \(0.8264097277 + 0.1252177048i\)
\(L(1)\) \(\approx\) \(0.8264097277 + 0.1252177048i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.753 + 0.657i)T \)
5 \( 1 + (0.936 + 0.351i)T \)
11 \( 1 + (0.858 + 0.512i)T \)
13 \( 1 + (0.753 - 0.657i)T \)
17 \( 1 + (-0.134 - 0.990i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.473 - 0.880i)T \)
29 \( 1 + (-0.691 - 0.722i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.691 - 0.722i)T \)
43 \( 1 + (-0.0448 - 0.998i)T \)
47 \( 1 + (-0.753 + 0.657i)T \)
53 \( 1 + (0.134 - 0.990i)T \)
59 \( 1 + (-0.0448 - 0.998i)T \)
61 \( 1 + (-0.473 + 0.880i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (0.222 + 0.974i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.753 - 0.657i)T \)
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

−17.86836110418875972512193528025, −17.12586141327647447204400737723, −16.82294206362116862029807453133, −16.21730249477455485117924041210, −15.29258673475880052587746533007, −14.48436905233084817904288789595, −13.55807940999245529604425843092, −13.30027000814850370187980928017, −12.53671462428712219436441743906, −11.75025557417374597451023860185, −11.143978924713592967908716570882, −10.569458112864315900159133324, −9.79342486925077962966322526310, −9.115448081476707698305956003216, −8.77783802458229451194371574888, −8.09194028703304637833997520130, −7.08471276832967935009763648861, −6.33436654490565299747501320963, −5.89753574261206639785007055310, −4.704182079140839158170140483540, −3.95755621963771241730584694880, −3.34709778863198193037087362099, −2.29909175935246319040659706132, −1.59436416856008669529389413127, −1.19081194759920904338767426117, 0.29271931343767180337237279869, 1.49048201739369199840843029593, 1.94396568460002154992651068455, 2.878210125380652101872929425416, 3.9429235713769298957772291008, 4.83891934840025866098431426336, 5.662214428161867855981507425238, 6.15028133297211610032321888557, 6.84993247756350230279328307095, 7.34619571281993920169158967607, 8.40857868099843937174219322264, 8.821241164812024439100124905176, 9.631647442151735386863846961349, 10.104076829674730204836032460145, 10.75795911339070601056110253387, 11.38051048170491110630015210522, 12.357461899880025824459702882890, 13.11929546083242823209335533434, 13.95029118547773713006449938804, 14.37285099283455968156023381424, 14.947649853125530405131776481485, 15.727432339119257730483735949935, 16.37177382594658099600562449769, 17.0274519437272165252055755195, 17.62564912171341804084813091994

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