↑

Properties

Label 1-967-967.50-r0-0-0
Degree $1$
Conductor $967$
Sign $0.677 + 0.735i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (−0.895 − 0.445i)2-s + (0.00975 + 0.999i)3-s + (0.602 + 0.797i)4-s + (−0.767 + 0.641i)5-s + (0.436 − 0.899i)6-s + (0.880 + 0.474i)7-s + (−0.184 − 0.982i)8-s + (−0.999 + 0.0195i)9-s + (0.972 − 0.232i)10-s + (−0.638 − 0.769i)11-s + (−0.791 + 0.610i)12-s + (−0.347 − 0.937i)13-s + (−0.576 − 0.816i)14-s + (−0.648 − 0.761i)15-s + (−0.272 + 0.962i)16-s + (0.527 − 0.849i)17-s + ⋯
L(s)  = 1  + (−0.895 − 0.445i)2-s + (0.00975 + 0.999i)3-s + (0.602 + 0.797i)4-s + (−0.767 + 0.641i)5-s + (0.436 − 0.899i)6-s + (0.880 + 0.474i)7-s + (−0.184 − 0.982i)8-s + (−0.999 + 0.0195i)9-s + (0.972 − 0.232i)10-s + (−0.638 − 0.769i)11-s + (−0.791 + 0.610i)12-s + (−0.347 − 0.937i)13-s + (−0.576 − 0.816i)14-s + (−0.648 − 0.761i)15-s + (−0.272 + 0.962i)16-s + (0.527 − 0.849i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.677 + 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7305047722 + 0.3201718785i\)
\(L(\frac12)\) \(\approx\) \(0.7305047722 + 0.3201718785i\)
\(L(1)\) \(\approx\) \(0.6527152061 + 0.1527756363i\)
\(L(1)\) \(\approx\) \(0.6527152061 + 0.1527756363i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.895 - 0.445i)T \)
3 \( 1 + (0.00975 + 0.999i)T \)
5 \( 1 + (-0.767 + 0.641i)T \)
7 \( 1 + (0.880 + 0.474i)T \)
11 \( 1 + (-0.638 - 0.769i)T \)
13 \( 1 + (-0.347 - 0.937i)T \)
17 \( 1 + (0.527 - 0.849i)T \)
19 \( 1 + (-0.996 + 0.0844i)T \)
23 \( 1 + (0.811 + 0.584i)T \)
29 \( 1 + (0.892 + 0.451i)T \)
31 \( 1 + (0.847 - 0.530i)T \)
37 \( 1 + (0.719 + 0.694i)T \)
41 \( 1 + (0.682 - 0.730i)T \)
43 \( 1 + (0.754 + 0.655i)T \)
47 \( 1 + (0.254 + 0.967i)T \)
53 \( 1 + (-0.715 - 0.699i)T \)
59 \( 1 + (0.341 - 0.940i)T \)
61 \( 1 + (0.291 + 0.956i)T \)
67 \( 1 + (-0.883 + 0.468i)T \)
71 \( 1 + (0.833 + 0.552i)T \)
73 \( 1 + (-0.715 + 0.699i)T \)
79 \( 1 + (0.840 + 0.541i)T \)
83 \( 1 + (-0.430 - 0.902i)T \)
89 \( 1 + (0.0357 - 0.999i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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

−21.19472628491272090114621207517, −20.69721072053457963642182839070, −19.65957665079909894778168704306, −19.34593292052512349792832970980, −18.49617623048056081646567764873, −17.62316588175407080795384082394, −17.05620299507057566685295885841, −16.46161011611506598245565354855, −15.24275024474060310526120714307, −14.70200341278776840186290871793, −13.8076928405483265940822395384, −12.63559512714293424760418541905, −12.043781424163202586778891936917, −11.128629151146992707158955005373, −10.42570180409200286176370227790, −9.11613046483759541790189626482, −8.35330188708063537328743104606, −7.80204697847819514198810542304, −7.143338990109184434723694641309, −6.28291539525342664819687316126, −5.065218708850967275453046525270, −4.32515686478367886394433993912, −2.520061729010200038631178248717, −1.647684075515650636577419637875, −0.72516208134518349246755947905, 0.77023502616260011250370869318, 2.74042344548032628509668321619, 2.86287600965530695973017962824, 4.116733566850800633523249748098, 5.09794789403091085113652905404, 6.183817760227634697702834019, 7.56574464237949527661616238413, 8.1184424477855811466147981414, 8.79381976707741804629497159757, 9.871607805574413344673539852648, 10.630682350167447649730491614077, 11.19561761203437894009044570559, 11.76062499397102572406351502114, 12.78811091965855741495670583162, 14.181410119992233753918610503011, 15.00576284854038644411188994247, 15.651094301259820182785475923451, 16.21719833242666742676894254536, 17.30698641036986767322228999889, 17.87364909043553493317515562018, 18.88954032924868977539095119722, 19.34061134323093208750001234220, 20.40595596429980584140188649783, 20.98058980334037783374129036484, 21.60223330881006826902349803257

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