↑

Properties

Label 1-837-837.301-r1-0-0
Degree $1$
Conductor $837$
Sign $0.909 + 0.414i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (−0.241 − 0.970i)2-s + (−0.882 + 0.469i)4-s + (0.766 − 0.642i)5-s + (−0.615 − 0.788i)7-s + (0.669 + 0.743i)8-s + (−0.809 − 0.587i)10-s + (−0.990 + 0.139i)11-s + (0.719 + 0.694i)13-s + (−0.615 + 0.788i)14-s + (0.559 − 0.829i)16-s + (−0.669 − 0.743i)17-s + (−0.809 − 0.587i)19-s + (−0.374 + 0.927i)20-s + (0.374 + 0.927i)22-s + (−0.990 − 0.139i)23-s + ⋯
L(s)  = 1  + (−0.241 − 0.970i)2-s + (−0.882 + 0.469i)4-s + (0.766 − 0.642i)5-s + (−0.615 − 0.788i)7-s + (0.669 + 0.743i)8-s + (−0.809 − 0.587i)10-s + (−0.990 + 0.139i)11-s + (0.719 + 0.694i)13-s + (−0.615 + 0.788i)14-s + (0.559 − 0.829i)16-s + (−0.669 − 0.743i)17-s + (−0.809 − 0.587i)19-s + (−0.374 + 0.927i)20-s + (0.374 + 0.927i)22-s + (−0.990 − 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.909 + 0.414i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.909 + 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4441047122 + 0.09647593077i\)
\(L(\frac12)\) \(\approx\) \(0.4441047122 + 0.09647593077i\)
\(L(1)\) \(\approx\) \(0.6227151246 - 0.4177635618i\)
\(L(1)\) \(\approx\) \(0.6227151246 - 0.4177635618i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.241 - 0.970i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (-0.615 - 0.788i)T \)
11 \( 1 + (-0.990 + 0.139i)T \)
13 \( 1 + (0.719 + 0.694i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.990 - 0.139i)T \)
29 \( 1 + (0.241 + 0.970i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.438 - 0.898i)T \)
43 \( 1 + (0.241 + 0.970i)T \)
47 \( 1 + (0.438 + 0.898i)T \)
53 \( 1 + (-0.669 - 0.743i)T \)
59 \( 1 + (-0.719 - 0.694i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (0.882 + 0.469i)T \)
83 \( 1 + (-0.961 - 0.275i)T \)
89 \( 1 + (0.978 + 0.207i)T \)
97 \( 1 + (-0.615 - 0.788i)T \)
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

−21.90089992236858860449186139130, −21.48541104876328807769382602537, −20.22712251679273258330956006666, −19.03419096027072117963774249980, −18.56342399729014124846764078646, −17.90271832563320631286512931194, −17.15484565261961098982516239363, −16.14957474466931098321947584100, −15.35908080059377764956193229637, −14.977997799771480301179736524307, −13.7268733359828640382318567413, −13.27238351769191502546085868434, −12.41368750283317939274008587151, −10.83955052901699964831042619984, −10.247180220817188212408696147196, −9.46448282883566549478101067629, −8.44519061043147393076101211683, −7.84561306058699875479516616474, −6.47090649780706483124333990824, −6.08140934626675415040376451745, −5.403080543852630275988847731881, −4.061324674659572599633582742923, −2.877204436613497641626521324004, −1.812855316284786581825342258089, −0.13172148263707734730105702823, 0.84922955589654842644174786612, 1.98502903250785647059313113272, 2.81980223112498422020062842307, 4.10820807989857463506921950690, 4.72208216224599963365649637077, 5.894158923418792547187295488115, 6.99718895975551993105271536256, 8.13680477464015670515595054937, 9.05565018002222965554269456729, 9.63308389847378714500668333521, 10.57509959707980539382042319718, 11.103299108544087814871442618939, 12.414789699998900008222148091379, 12.99880350088185070757381884627, 13.63202370978842203759715284215, 14.26266517452441485861702081497, 15.99652814368084002123448534901, 16.367941241831048166223228687266, 17.50792570491597618915737948625, 17.94826241327585169746210986179, 18.87880538802769318883302605626, 19.80424698867761215790528041499, 20.423191776346026051919361607258, 21.06504969759862560977479894140, 21.76255228034874716206716305657

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