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Properties

Label 1-800-800.213-r1-0-0
Degree $1$
Conductor $800$
Sign $-0.389 - 0.920i$
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 + 7-s + (0.951 + 0.309i)9-s + (−0.453 + 0.891i)11-s + (−0.891 + 0.453i)13-s + (0.587 − 0.809i)17-s + (−0.987 + 0.156i)19-s + (−0.987 − 0.156i)21-s + (0.309 + 0.951i)23-s + (−0.891 − 0.453i)27-s + (0.156 − 0.987i)29-s + (−0.809 − 0.587i)31-s + (0.587 − 0.809i)33-s + (0.453 + 0.891i)37-s + (0.951 − 0.309i)39-s + ⋯
L(s)  = 1  + (−0.987 − 0.156i)3-s + 7-s + (0.951 + 0.309i)9-s + (−0.453 + 0.891i)11-s + (−0.891 + 0.453i)13-s + (0.587 − 0.809i)17-s + (−0.987 + 0.156i)19-s + (−0.987 − 0.156i)21-s + (0.309 + 0.951i)23-s + (−0.891 − 0.453i)27-s + (0.156 − 0.987i)29-s + (−0.809 − 0.587i)31-s + (0.587 − 0.809i)33-s + (0.453 + 0.891i)37-s + (0.951 − 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.389 - 0.920i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ -0.389 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3298275045 - 0.4978428962i\)
\(L(\frac12)\) \(\approx\) \(0.3298275045 - 0.4978428962i\)
\(L(1)\) \(\approx\) \(0.7407112424 + 0.02037599205i\)
\(L(1)\) \(\approx\) \(0.7407112424 + 0.02037599205i\)

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 \( 1 + T \)
11 \( 1 + (-0.453 + 0.891i)T \)
13 \( 1 + (-0.891 + 0.453i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.987 + 0.156i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.156 - 0.987i)T \)
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 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (0.156 - 0.987i)T \)
59 \( 1 + (0.891 - 0.453i)T \)
61 \( 1 + (0.891 + 0.453i)T \)
67 \( 1 + (0.156 + 0.987i)T \)
71 \( 1 + (-0.587 - 0.809i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.156 + 0.987i)T \)
89 \( 1 + (0.951 - 0.309i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

−22.11315046442860701834481298237, −21.61718673565990486216238912634, −21.00449464054881004572470142065, −19.95714898310121420653724332839, −18.88474840208709344589454050134, −18.246730001742720247984056646854, −17.370724660342901371720259495597, −16.82222126659365106119965092628, −16.02864255182890973293493816178, −14.91855629099318560548048703549, −14.47202536032626266593397264926, −13.081801690219896546417946024592, −12.463270178706837963897866748472, −11.578935799144422166090461080728, −10.58972814880665268055137206293, −10.4518876250769371513324945280, −8.945818834330518841367860038, −8.09388301679020733344044832379, −7.160641061157238607633080807648, −6.11714713112887119527950971198, −5.26267697568615325266775864139, −4.65237973353993213919084051922, −3.47375550744785590120863605569, −2.091840182125452258212140168051, −0.92969694414855379927670907033, 0.180076721805418533036585562051, 1.54069962482209210597062475485, 2.34573031258763428808795363154, 4.066196641963906216766315431204, 4.93173545565619089337614875332, 5.42878131540355637877764693768, 6.743239828224113236033664479640, 7.41321604046389173688744739534, 8.234191779600190688877457530629, 9.69924985989902353302517449679, 10.17468403835958621733204421491, 11.454844216758468027519286480926, 11.68850403500377985892192916963, 12.71318228560640594604277740289, 13.51130125904120370536487474858, 14.721184688931196273010529324031, 15.22512444247200653074817001127, 16.381991286847658136129503715285, 17.118442946728579303529098715867, 17.68601221689186997642923593579, 18.46975794369239896377835404263, 19.21202674681076693339538986594, 20.38570794358292550822308581641, 21.15902404695196887060943685720, 21.77825496449611287119187250842

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