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Properties

Label 1-800-800.571-r1-0-0
Degree $1$
Conductor $800$
Sign $0.331 + 0.943i$
Analytic cond. $85.9719$
Root an. cond. $85.9719$
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  + (0.891 + 0.453i)3-s − i·7-s + (0.587 + 0.809i)9-s + (−0.156 − 0.987i)11-s + (−0.156 + 0.987i)13-s + (−0.309 + 0.951i)17-s + (0.453 + 0.891i)19-s + (0.453 − 0.891i)21-s + (0.587 − 0.809i)23-s + (0.156 + 0.987i)27-s + (−0.891 − 0.453i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.987 − 0.156i)37-s + (−0.587 + 0.809i)39-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)3-s − i·7-s + (0.587 + 0.809i)9-s + (−0.156 − 0.987i)11-s + (−0.156 + 0.987i)13-s + (−0.309 + 0.951i)17-s + (0.453 + 0.891i)19-s + (0.453 − 0.891i)21-s + (0.587 − 0.809i)23-s + (0.156 + 0.987i)27-s + (−0.891 − 0.453i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.987 − 0.156i)37-s + (−0.587 + 0.809i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.331 + 0.943i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ 0.331 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.199580759 + 1.558831323i\)
\(L(\frac12)\) \(\approx\) \(2.199580759 + 1.558831323i\)
\(L(1)\) \(\approx\) \(1.412057448 + 0.2807674047i\)
\(L(1)\) \(\approx\) \(1.412057448 + 0.2807674047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.891 + 0.453i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.156 - 0.987i)T \)
13 \( 1 + (-0.156 + 0.987i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.453 + 0.891i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (-0.891 - 0.453i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.987 - 0.156i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.453 - 0.891i)T \)
59 \( 1 + (0.987 + 0.156i)T \)
61 \( 1 + (0.987 - 0.156i)T \)
67 \( 1 + (-0.453 - 0.891i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.453 + 0.891i)T \)
89 \( 1 + (0.587 - 0.809i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−22.00827408276867536048624970188, −20.76187431368048030295800726202, −20.42889834729269161267315708642, −19.50725993203313113958872739453, −18.72650011063424133357764305649, −17.9647979892852009206427594061, −17.44094043914507554628256740533, −15.87215109018807107384116361645, −15.308002648203912450494569044102, −14.78690872870277156785914645774, −13.65046299089718927551063112040, −12.969929158683942289691687912467, −12.24615731066052161888983174032, −11.39028446165099202551288766725, −10.089252625261022595549747335142, −9.23642713816382804444933246107, −8.73937274923977449344051417411, −7.43027624665573343550094123848, −7.18418139072319097195082490541, −5.71620748958615419910819756361, −4.92818796080096314288616227783, −3.58590330578860012280032741313, −2.63293918567079986339067735644, −1.991586627186540340027888665755, −0.55854110029050458447927695952, 1.09627598543192780676800065772, 2.19547860435299880628908450477, 3.45207708116287561487541108682, 3.98664975938533474719351750094, 4.983997962377627187833749486857, 6.27167689023155131386968615342, 7.26194861411716928553999972970, 8.12181446610950614201256515997, 8.8723955461463298595412024777, 9.78752452553155491101359459931, 10.6334306496491744459159198806, 11.254565669740959427062052010706, 12.66009621989554311020188805193, 13.41353486358501230545564870026, 14.27114451839182017369717032080, 14.60423096501994509581917088393, 15.91106818765645464747584133379, 16.45159351374721119488630429171, 17.16513680854364906998235440146, 18.44359785353050402570121524593, 19.27259412849529063599667694526, 19.69851223137951750794587940200, 20.88648943292325519922418740095, 21.08157382377785727688141330029, 22.11656704112328821792430452606

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