↑

Properties

Label 1-60e2-3600.133-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.905 + 0.424i$
Analytic cond. $386.873$
Root an. cond. $386.873$
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  + (0.866 − 0.5i)7-s + (−0.406 + 0.913i)11-s + (0.913 − 0.406i)13-s + (0.951 − 0.309i)17-s + (−0.951 + 0.309i)19-s + (−0.994 + 0.104i)23-s + (−0.207 − 0.978i)29-s + (−0.978 − 0.207i)31-s + (−0.809 − 0.587i)37-s + (−0.913 + 0.406i)41-s + (0.5 + 0.866i)43-s + (0.207 + 0.978i)47-s + (0.5 − 0.866i)49-s + (−0.309 + 0.951i)53-s + (−0.406 − 0.913i)59-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (−0.406 + 0.913i)11-s + (0.913 − 0.406i)13-s + (0.951 − 0.309i)17-s + (−0.951 + 0.309i)19-s + (−0.994 + 0.104i)23-s + (−0.207 − 0.978i)29-s + (−0.978 − 0.207i)31-s + (−0.809 − 0.587i)37-s + (−0.913 + 0.406i)41-s + (0.5 + 0.866i)43-s + (0.207 + 0.978i)47-s + (0.5 − 0.866i)49-s + (−0.309 + 0.951i)53-s + (−0.406 − 0.913i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.905 + 0.424i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.905 + 0.424i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.243363000 + 0.4993959233i\)
\(L(\frac12)\) \(\approx\) \(2.243363000 + 0.4993959233i\)
\(L(1)\) \(\approx\) \(1.155029352 + 0.002649209294i\)
\(L(1)\) \(\approx\) \(1.155029352 + 0.002649209294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.406 + 0.913i)T \)
13 \( 1 + (0.913 - 0.406i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 + (-0.994 + 0.104i)T \)
29 \( 1 + (-0.207 - 0.978i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.207 + 0.978i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.406 - 0.913i)T \)
61 \( 1 + (0.406 - 0.913i)T \)
67 \( 1 + (0.978 + 0.207i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.207 + 0.978i)T \)
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

−18.39139784675198989022945232381, −18.03938106115294925182069611360, −16.953436395782787454793059423798, −16.51962487994072983869914569113, −15.682860601712726988521353257954, −15.0715911679137348161340293689, −14.25172102020857722454271463877, −13.801340739057336285509497916253, −12.93917693237292336446432063399, −12.14503088750644891093352969767, −11.54838296968275703009823548242, −10.7192697668916521875496232288, −10.37832969170603689210884603149, −9.09628992249598327308865040112, −8.56706984936188533805536449491, −8.102947270630260242498374021349, −7.15074284561051549511406596681, −6.238539399440324996507379358728, −5.56570395377407774872635500403, −4.97541010261573583985511198752, −3.86320015171811713942228148477, −3.33932545778307677265939244125, −2.134352580391198931338140, −1.579373629739051296369447412888, −0.46627926400568545911982764527, 0.65712160554045422376069944295, 1.63742050840127741852079075978, 2.24268241109152013287775215742, 3.47009835058614362615632935149, 4.113315801672447894015921687015, 4.89217231950051509518663299325, 5.6756298764093440691303617948, 6.4191723513060884325322950309, 7.49821465801855641636135471611, 7.8617031249479874522055084597, 8.558956291714517698533310751956, 9.61070684544599235709421283708, 10.20683079238929127543583260083, 10.9176558096978675195447310816, 11.51143123036640665324905389761, 12.47336393736862291843910759011, 12.91052068817822593848781486253, 13.94058676374755531567603077681, 14.319315252152845843328256863, 15.17478342235471340348067509058, 15.737409591779743968572468651567, 16.58528202663594658544338303638, 17.2929904063047651887703007170, 17.851926306303981621492392434070, 18.49433942447346575572597529432

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