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Properties

Label 1-837-837.220-r1-0-0
Degree $1$
Conductor $837$
Sign $0.451 + 0.892i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.848 − 0.529i)2-s + (0.438 − 0.898i)4-s + (0.173 + 0.984i)5-s + (−0.241 − 0.970i)7-s + (−0.104 − 0.994i)8-s + (0.669 + 0.743i)10-s + (0.241 + 0.970i)11-s + (−0.0348 + 0.999i)13-s + (−0.719 − 0.694i)14-s + (−0.615 − 0.788i)16-s + (0.809 − 0.587i)17-s + (−0.978 + 0.207i)19-s + (0.961 + 0.275i)20-s + (0.719 + 0.694i)22-s + (0.719 + 0.694i)23-s + ⋯
L(s)  = 1  + (0.848 − 0.529i)2-s + (0.438 − 0.898i)4-s + (0.173 + 0.984i)5-s + (−0.241 − 0.970i)7-s + (−0.104 − 0.994i)8-s + (0.669 + 0.743i)10-s + (0.241 + 0.970i)11-s + (−0.0348 + 0.999i)13-s + (−0.719 − 0.694i)14-s + (−0.615 − 0.788i)16-s + (0.809 − 0.587i)17-s + (−0.978 + 0.207i)19-s + (0.961 + 0.275i)20-s + (0.719 + 0.694i)22-s + (0.719 + 0.694i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.451 + 0.892i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (220, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.451 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.100124549 + 1.291125018i\)
\(L(\frac12)\) \(\approx\) \(2.100124549 + 1.291125018i\)
\(L(1)\) \(\approx\) \(1.570529381 - 0.1495494925i\)
\(L(1)\) \(\approx\) \(1.570529381 - 0.1495494925i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.848 - 0.529i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.241 - 0.970i)T \)
11 \( 1 + (0.241 + 0.970i)T \)
13 \( 1 + (-0.0348 + 0.999i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.719 + 0.694i)T \)
29 \( 1 + (-0.848 + 0.529i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.615 + 0.788i)T \)
43 \( 1 + (0.882 + 0.469i)T \)
47 \( 1 + (0.990 - 0.139i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (0.848 + 0.529i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.559 + 0.829i)T \)
83 \( 1 + (0.882 + 0.469i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (0.961 + 0.275i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.931085176158035624294667613822, −21.031230486296981956771581824305, −20.64561003102208513402182031971, −19.42687194297982859655721634295, −18.70722688945158381250511144764, −17.2923027450754841799445853715, −17.01804814426485207580890875226, −15.99773937529268298315979592488, −15.4139438725649464326979142911, −14.59411381341239339195406601990, −13.632399830820674292348195785168, −12.6937395912728401633774494453, −12.49769967601668385439124912849, −11.44371831416992380792609895564, −10.4026855226145266131855104586, −8.94968350611520501658453892488, −8.56825119896812032929879658383, −7.67038212651859021940838062302, −6.30503940504115245358526970862, −5.70488386853669299781522232002, −5.08382818388318706851556699537, −3.92188436477643029166806398667, −3.01643629392514478004535791160, −1.92318424050741406867245028643, −0.37323784983094557596696720125, 1.29930265403110570836383258259, 2.20595253200303051757642232747, 3.32346776886182157044456790247, 4.021552440404095235267801926364, 4.95393494434486268931520000910, 6.1435767527416279447845811939, 6.99472505739565407403259743516, 7.39138117628305726332239511324, 9.28415679867175176420314394706, 9.9679195640773447763703057264, 10.70651946502541701738064697668, 11.43785725481845151064124966712, 12.32339184256049990855223457105, 13.25313921015465983079556201348, 14.02507780682463489736337467280, 14.6006048769873619139002785947, 15.29732680616497326363024120868, 16.42281507363649715618801907278, 17.238502983080954130090001798200, 18.32668912052114981412694742412, 19.1507034410108477654413261037, 19.65072768262457348839268479246, 20.7522847668102464463755651540, 21.18892650052122018154015574660, 22.24099198299080712281358248020

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