↑

Properties

Label 1-6027-6027.860-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.987 - 0.159i$
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  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s + (0.900 + 0.433i)8-s + (0.900 − 0.433i)10-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.900 + 0.433i)16-s + (0.222 − 0.974i)17-s + 19-s + (−0.222 + 0.974i)20-s + (0.222 + 0.974i)22-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (0.222 + 0.974i)26-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s + (0.900 + 0.433i)8-s + (0.900 − 0.433i)10-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.900 + 0.433i)16-s + (0.222 − 0.974i)17-s + 19-s + (−0.222 + 0.974i)20-s + (0.222 + 0.974i)22-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (0.222 + 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.987 - 0.159i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (860, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.987 - 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003454013455 - 0.04300599706i\)
\(L(\frac12)\) \(\approx\) \(0.003454013455 - 0.04300599706i\)
\(L(1)\) \(\approx\) \(0.6167846517 + 0.07375793869i\)
\(L(1)\) \(\approx\) \(0.6167846517 + 0.07375793869i\)

Euler product

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

−18.33538993405917922075444119212, −17.42742117779905277291122201438, −16.7880981577301731563639824392, −16.21203542558633298363336237318, −15.509508833619501376348273916132, −14.69402942770795336337001187922, −14.15338632871702234286633153553, −13.19493728578657687382753605157, −12.569507962867652058857366322328, −11.79408502107449409723966144783, −11.5842092524881722218654752195, −10.69397291121853406872161878193, −10.22370754782554343965722141495, −9.34081453615390536386579371446, −8.78228366867223197821879600408, −8.093642172629498108442186035243, −7.3270205134223878434402854809, −6.89620918944046322355538410992, −5.98321765899328026822401705741, −4.74375272656935164147361017098, −4.01625655275699859052379506270, −3.68343435538408578879537428587, −2.77392110615607173471199077483, −1.87868781096708980859653118690, −1.21814541759691317803379391979, 0.01608719720250031630959954247, 1.06456084894256363248560295441, 1.493293696897519830892678212189, 3.23521488671794114209867614363, 3.43851518931449353974319603597, 4.71111681741702466816608407731, 5.20623760856708146947281255341, 5.90798263720534541152442593485, 6.74761089447259939111818025769, 7.51360578637403789312762882812, 7.89686075025390513097211229797, 8.73231379454884793621829892527, 9.16702067260830794579919344773, 9.87591604706842543079019847010, 10.844415945741029745970972049377, 11.377432524755168033032441095632, 11.89846680998494702546430867961, 12.98616840570113317718805154889, 13.55288757602834008825761029695, 14.285652264093364223523423064132, 14.95259909972793029196631451886, 15.64358822122389483946392244185, 16.19341957072184269328132986684, 16.515228149882652931564891538570, 17.31389026502425204503641864551

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