↑

Properties

Label 1-927-927.286-r1-0-0
Degree $1$
Conductor $927$
Sign $0.0445 + 0.999i$
Analytic cond. $99.6199$
Root an. cond. $99.6199$
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  + (0.992 + 0.122i)2-s + (0.969 + 0.243i)4-s + (0.908 + 0.417i)5-s + (−0.779 − 0.626i)7-s + (0.932 + 0.361i)8-s + (0.850 + 0.526i)10-s + (−0.992 − 0.122i)11-s + (−0.153 + 0.988i)13-s + (−0.696 − 0.717i)14-s + (0.881 + 0.473i)16-s + (−0.982 + 0.183i)17-s + (0.445 − 0.895i)19-s + (0.779 + 0.626i)20-s + (−0.969 − 0.243i)22-s + (0.992 − 0.122i)23-s + ⋯
L(s)  = 1  + (0.992 + 0.122i)2-s + (0.969 + 0.243i)4-s + (0.908 + 0.417i)5-s + (−0.779 − 0.626i)7-s + (0.932 + 0.361i)8-s + (0.850 + 0.526i)10-s + (−0.992 − 0.122i)11-s + (−0.153 + 0.988i)13-s + (−0.696 − 0.717i)14-s + (0.881 + 0.473i)16-s + (−0.982 + 0.183i)17-s + (0.445 − 0.895i)19-s + (0.779 + 0.626i)20-s + (−0.969 − 0.243i)22-s + (0.992 − 0.122i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0445 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0445 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.0445 + 0.999i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.0445 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.922866093 + 2.795457530i\)
\(L(\frac12)\) \(\approx\) \(2.922866093 + 2.795457530i\)
\(L(1)\) \(\approx\) \(1.964195276 + 0.5302668950i\)
\(L(1)\) \(\approx\) \(1.964195276 + 0.5302668950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.992 + 0.122i)T \)
5 \( 1 + (0.908 + 0.417i)T \)
7 \( 1 + (-0.779 - 0.626i)T \)
11 \( 1 + (-0.992 - 0.122i)T \)
13 \( 1 + (-0.153 + 0.988i)T \)
17 \( 1 + (-0.982 + 0.183i)T \)
19 \( 1 + (0.445 - 0.895i)T \)
23 \( 1 + (0.992 - 0.122i)T \)
29 \( 1 + (0.816 - 0.577i)T \)
31 \( 1 + (0.0307 + 0.999i)T \)
37 \( 1 + (-0.739 + 0.673i)T \)
41 \( 1 + (0.816 + 0.577i)T \)
43 \( 1 + (-0.213 + 0.976i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.445 + 0.895i)T \)
59 \( 1 + (-0.779 + 0.626i)T \)
61 \( 1 + (0.650 + 0.759i)T \)
67 \( 1 + (0.779 - 0.626i)T \)
71 \( 1 + (-0.0922 + 0.995i)T \)
73 \( 1 + (-0.0922 + 0.995i)T \)
79 \( 1 + (0.816 - 0.577i)T \)
83 \( 1 + (-0.779 - 0.626i)T \)
89 \( 1 + (0.273 - 0.961i)T \)
97 \( 1 + (0.332 + 0.943i)T \)
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

−21.53470089131319596492282287241, −20.75957197535862824732339522308, −20.256652139021329290262455989379, −19.25030921784170071810890494614, −18.34939508130465671119240809560, −17.481287867788509314927041294600, −16.48136102705301345077091662149, −15.71371301622309794293340563572, −15.18900184476207076365553607392, −14.0756246300591039734429640762, −13.31163569526025008944530827831, −12.74623616736302988146348250310, −12.21614001111267325871244740904, −10.90695583860534517102520587830, −10.22821288735789376998031944745, −9.41634269585482220677667485142, −8.325620695418550688860951969704, −7.17260943277091557913626402757, −6.25808230866282450089928658781, −5.41591768516693292403539709323, −5.03867371113977313815984178633, −3.605374528153038687099836693003, −2.6713289730611183118086536675, −2.03128466357779315353850989370, −0.55563376027572521743250259410, 1.26233812120125260336505014221, 2.59561353562861437355739325405, 2.98744761289109731389260644902, 4.35469957582237267854190686580, 5.04961295897424378392383146165, 6.19622719416336103932204685573, 6.74197846830583299535295356415, 7.4320002361369018345799791235, 8.8392736746442373728549943926, 9.841781158893774775888948687937, 10.69080361050547443225770921915, 11.27001461223530416645532911804, 12.52746907322939632681577800621, 13.28930856206876038014598190279, 13.688063012209567648631764724912, 14.45848717919764203248119044469, 15.53370050293685078951485224106, 16.07497345736405559794556451905, 17.0488974256266618756164776168, 17.68391957086136568960316613152, 18.862760014194107239428623569605, 19.62426541915290860852587840778, 20.48462982710596378421758709956, 21.3837876230016758467295846863, 21.746221380776975621942899517076

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