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Properties

Label 1-837-837.29-r0-0-0
Degree $1$
Conductor $837$
Sign $0.853 + 0.520i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (−0.0348 + 0.999i)2-s + (−0.997 − 0.0697i)4-s + (−0.173 + 0.984i)5-s + (−0.241 + 0.970i)7-s + (0.104 − 0.994i)8-s + (−0.978 − 0.207i)10-s + (−0.719 − 0.694i)11-s + (−0.0348 − 0.999i)13-s + (−0.961 − 0.275i)14-s + (0.990 + 0.139i)16-s + (0.913 − 0.406i)17-s + (0.669 − 0.743i)19-s + (0.241 − 0.970i)20-s + (0.719 − 0.694i)22-s + (−0.241 − 0.970i)23-s + ⋯
L(s)  = 1  + (−0.0348 + 0.999i)2-s + (−0.997 − 0.0697i)4-s + (−0.173 + 0.984i)5-s + (−0.241 + 0.970i)7-s + (0.104 − 0.994i)8-s + (−0.978 − 0.207i)10-s + (−0.719 − 0.694i)11-s + (−0.0348 − 0.999i)13-s + (−0.961 − 0.275i)14-s + (0.990 + 0.139i)16-s + (0.913 − 0.406i)17-s + (0.669 − 0.743i)19-s + (0.241 − 0.970i)20-s + (0.719 − 0.694i)22-s + (−0.241 − 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.853 + 0.520i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.853 + 0.520i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9096566300 + 0.2552959775i\)
\(L(\frac12)\) \(\approx\) \(0.9096566300 + 0.2552959775i\)
\(L(1)\) \(\approx\) \(0.7510928642 + 0.3953360512i\)
\(L(1)\) \(\approx\) \(0.7510928642 + 0.3953360512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.0348 + 0.999i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.241 + 0.970i)T \)
11 \( 1 + (-0.719 - 0.694i)T \)
13 \( 1 + (-0.0348 - 0.999i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.241 - 0.970i)T \)
29 \( 1 + (0.0348 - 0.999i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.615 + 0.788i)T \)
43 \( 1 + (0.882 - 0.469i)T \)
47 \( 1 + (0.374 + 0.927i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.882 + 0.469i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (-0.559 - 0.829i)T \)
83 \( 1 + (0.0348 - 0.999i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (-0.719 - 0.694i)T \)
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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

−21.862595298326129392153182149130, −20.972824318748126898655627765622, −20.560449901792746703893961720406, −19.81100128281519412420123643831, −19.12043628991077276311531404558, −18.202447991838886334279075524204, −17.26375106354525778325289621452, −16.65684698575290964903449674457, −15.837045281418441242577143725786, −14.46738577203403200063783995470, −13.77373750226910279408196538511, −12.959139032774184284895888421690, −12.30982323468726731560965504261, −11.57186737795776769794840096538, −10.49194831910964082311153108761, −9.801226820409946979983926655113, −9.12954182169981993664347589929, −7.97197784214657511037413552931, −7.38958568286965706080092874312, −5.763930376061440885223754755599, −4.84150925380995060984807141537, −4.08533395697279400339826681613, −3.279024013303906144817610478174, −1.80843221831713021133483466531, −1.08679922852267521281780993381, 0.52369837191102933382259453163, 2.669642252848584406693051744966, 3.19868471822001012197328674266, 4.57871276905778751187916311558, 5.838905484166320627956642329759, 5.9362075548324351253960184450, 7.33711130896971468340799663562, 7.85543683631348991259234786745, 8.80284673849512330211878720870, 9.78224920279028354087204818108, 10.54044925522063281702309329576, 11.62070374857800173746131039958, 12.64850886140140819202739579605, 13.46146229236161676397286150361, 14.401737655312992121668870092791, 14.99811837542955268652040648939, 15.870439021124830287402180582070, 16.19866644919722816496943684551, 17.6362663197102344558769023914, 18.10685655340903606175221400954, 18.84814832720367540520378517668, 19.37330826099120394002480078199, 20.80807055821463655060362497574, 21.74214214429120171470286703450, 22.42721951988901599360712827255

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