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Properties

Label 1-800-800.411-r1-0-0
Degree $1$
Conductor $800$
Sign $0.712 - 0.701i$
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  + (−0.987 + 0.156i)3-s − i·7-s + (0.951 − 0.309i)9-s + (0.891 − 0.453i)11-s + (0.891 + 0.453i)13-s + (0.809 − 0.587i)17-s + (0.156 − 0.987i)19-s + (0.156 + 0.987i)21-s + (0.951 + 0.309i)23-s + (−0.891 + 0.453i)27-s + (0.987 − 0.156i)29-s + (0.809 − 0.587i)31-s + (−0.809 + 0.587i)33-s + (−0.453 + 0.891i)37-s + (−0.951 − 0.309i)39-s + ⋯
L(s)  = 1  + (−0.987 + 0.156i)3-s − i·7-s + (0.951 − 0.309i)9-s + (0.891 − 0.453i)11-s + (0.891 + 0.453i)13-s + (0.809 − 0.587i)17-s + (0.156 − 0.987i)19-s + (0.156 + 0.987i)21-s + (0.951 + 0.309i)23-s + (−0.891 + 0.453i)27-s + (0.987 − 0.156i)29-s + (0.809 − 0.587i)31-s + (−0.809 + 0.587i)33-s + (−0.453 + 0.891i)37-s + (−0.951 − 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.712 - 0.701i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ 0.712 - 0.701i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.830545196 - 0.7498283892i\)
\(L(\frac12)\) \(\approx\) \(1.830545196 - 0.7498283892i\)
\(L(1)\) \(\approx\) \(1.014021504 - 0.1566547975i\)
\(L(1)\) \(\approx\) \(1.014021504 - 0.1566547975i\)

Euler product

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

−22.41862848176906732999191532890, −21.27904919504666802099809543206, −20.959786333825588774071172109512, −19.46960026442676344146677539765, −18.92774159038307467643140660261, −18.00231074316900104961179176054, −17.49611340311716301225097896460, −16.49192493468728686646284613474, −15.8603777181683356107049134958, −14.95404193261191389508785326518, −14.106318378436867351852800108220, −12.77411284822233166352106686671, −12.36508919333973601512996697464, −11.60548964068284654054798528779, −10.688849145347113757233412847910, −9.88553615803896282053452696817, −8.84380579100879855709155658649, −7.94925117554188023749513881914, −6.77372867553603603959077773492, −6.03690221535104899910192771278, −5.36946708496822383214621580339, −4.29857883604594717363007653398, −3.199559390763540919091200660802, −1.75827892250134104981664642473, −0.92353669609866924223051433141, 0.77244229916545668045771964719, 1.203994591962540557624431679936, 3.09565905099727019163145410627, 4.10784502772968175491863653880, 4.81785952546738334164064338256, 5.98017905706376955112753083747, 6.71868496161525906666161913825, 7.44138892387505373200235083682, 8.75277454798113686305817974897, 9.68210092391596580278009831542, 10.511328165472946079937166855044, 11.413320404505998266675081617529, 11.76167833493890089870491609154, 13.07644177518776941683217409281, 13.6983734777289457351871915993, 14.617203625907526126663643173279, 15.84355211854754097474867631222, 16.33254023310589610248927525679, 17.2203763044081953836108193001, 17.64014657431942679725178850206, 18.810316817766924665828711193606, 19.406450015537143325021465310863, 20.578423705713436287632876749587, 21.19181329783670384527934410632, 22.04048284732267284357018894666

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