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Properties

Label 1-800-800.171-r1-0-0
Degree $1$
Conductor $800$
Sign $0.943 - 0.331i$
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

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

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ 0.943 - 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.332161154 - 0.2271122724i\)
\(L(\frac12)\) \(\approx\) \(1.332161154 - 0.2271122724i\)
\(L(1)\) \(\approx\) \(0.8265681351 - 0.1477280770i\)
\(L(1)\) \(\approx\) \(0.8265681351 - 0.1477280770i\)

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

−22.00501766802900452390196492451, −21.392078123580601211322437949911, −20.92861650735356082540877039154, −19.56305562911842425943137302860, −18.66708156659474724764436308507, −18.24968183196548389087144751639, −17.11403590810010059373705729523, −16.45015705827191157319055982999, −15.79557280145633585513151145804, −14.99113725382508292270177536400, −14.01637731944636447633452903216, −13.02624561108471362074317188175, −11.93940987126588199123809020549, −11.54246139021300805798688119750, −10.75474946936213916658489565495, −9.51886442838769764096112420090, −9.08928512660776354956446194807, −7.93571510886998768151817288039, −6.61458024322283122890682187704, −6.013242508444409315554204102619, −5.1814559236587951581626752467, −4.223983302501845357314613688301, −3.18339493644735478674020888237, −1.892816577692832211791968804444, −0.55747594467472382317854987543, 0.668265651117311485206532918753, 1.548220570628499368952660515057, 2.87736042782550801996013243829, 4.33574787143746762444126368072, 4.83499023234311689396587632498, 6.11568766225871731977543371804, 6.83437225506377059068337715582, 7.55944576069678615910346971386, 8.533445832614093707933868229611, 9.88706424375880844268507505800, 10.66674666146802362801771087850, 11.09289282548671213496014907705, 12.50473954769666359812193885977, 12.76009103096654012881335933416, 13.69168702465690605924968984907, 14.77815323861857932616603536586, 15.61969869145449251487265242169, 16.59492915155456858587835557137, 17.401946753476355764597681724816, 17.70358428315782066417042879709, 18.69022117088311952238807837049, 19.83029824329754939413831940873, 20.12398009175600119961931874768, 21.40072767006873105608888071772, 22.13234039151971912713122494168

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