↑

Properties

Label 1-828-828.211-r1-0-0
Degree $1$
Conductor $828$
Sign $-0.885 - 0.464i$
Analytic cond. $88.9809$
Root an. cond. $88.9809$
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  + (0.580 − 0.814i)5-s + (−0.235 + 0.971i)7-s + (−0.981 + 0.189i)11-s + (0.235 + 0.971i)13-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.327 − 0.945i)25-s + (−0.786 − 0.618i)29-s + (−0.0475 + 0.998i)31-s + (0.654 + 0.755i)35-s + (0.415 − 0.909i)37-s + (0.580 − 0.814i)41-s + (−0.0475 − 0.998i)43-s + (0.5 − 0.866i)47-s + (−0.888 − 0.458i)49-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)5-s + (−0.235 + 0.971i)7-s + (−0.981 + 0.189i)11-s + (0.235 + 0.971i)13-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.327 − 0.945i)25-s + (−0.786 − 0.618i)29-s + (−0.0475 + 0.998i)31-s + (0.654 + 0.755i)35-s + (0.415 − 0.909i)37-s + (0.580 − 0.814i)41-s + (−0.0475 − 0.998i)43-s + (0.5 − 0.866i)47-s + (−0.888 − 0.458i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.885 - 0.464i$
Analytic conductor: \(88.9809\)
Root analytic conductor: \(88.9809\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 828,\ (1:\ ),\ -0.885 - 0.464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02978131560 - 0.1209965556i\)
\(L(\frac12)\) \(\approx\) \(0.02978131560 - 0.1209965556i\)
\(L(1)\) \(\approx\) \(0.9117892087 + 0.05020862662i\)
\(L(1)\) \(\approx\) \(0.9117892087 + 0.05020862662i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.580 - 0.814i)T \)
7 \( 1 + (-0.235 + 0.971i)T \)
11 \( 1 + (-0.981 + 0.189i)T \)
13 \( 1 + (0.235 + 0.971i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (-0.786 - 0.618i)T \)
31 \( 1 + (-0.0475 + 0.998i)T \)
37 \( 1 + (0.415 - 0.909i)T \)
41 \( 1 + (0.580 - 0.814i)T \)
43 \( 1 + (-0.0475 - 0.998i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (-0.235 - 0.971i)T \)
61 \( 1 + (-0.888 + 0.458i)T \)
67 \( 1 + (-0.981 - 0.189i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (-0.723 + 0.690i)T \)
83 \( 1 + (-0.580 - 0.814i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.995 - 0.0950i)T \)
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

−22.442112188636039090829451545745, −21.55863275491674912357625782984, −20.59363749282218029402511314754, −20.113805393345749249862769423870, −19.00084490196044947116810998975, −18.19668169277328842892558717217, −17.68768600604736565765439429450, −16.734222532286832963550207652888, −15.83101087558260604546611880296, −15.06084398392912774935077087274, −14.106177906697321945598880650484, −13.31204841923078903054777312853, −12.95222904760731768027939477880, −11.30289279456837329007417507412, −10.87678225768712506412875972406, −10.00331447344708967839367903407, −9.33315470781396717050029710881, −7.89539036085670528212768893475, −7.33579072376879198954886378642, −6.3874356665321571963851473476, −5.47897328433908074915860196532, −4.477872625429700240674027728221, −3.11432492005877189136629182011, −2.687630464890483020123935654, −1.12869571230812987465244472240, 0.027140135560937390366961456912, 1.66860622500886598121012149236, 2.245269422311285412982292218318, 3.63474316965249394590115750813, 4.718818312720051430624170389550, 5.669122790722748747412843572003, 6.1636362689171974281678686314, 7.52736843576548829568617834540, 8.52946379573944867442367104467, 9.112002059743073148598736174979, 9.986475035686551932518283277444, 10.88872831743751082572934146569, 12.13509478641545913106915863617, 12.56185122815706167721929392262, 13.422376037804797209687126957066, 14.2841909318582564965812267970, 15.30518190413340930272166668243, 16.04366682406897718929676077420, 16.749333697131068580680964157620, 17.64622535392694975658816315136, 18.49135232212433314193378988429, 19.102620930841761288546354650130, 20.14593928525273730934206074391, 21.11973979889813086639362754701, 21.362907785174302260179863592010

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