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Properties

Label 1-800-800.779-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: $\chi_{800} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ -0.943 + 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3799684810 + 2.228762210i\)
\(L(\frac12)\) \(\approx\) \(0.3799684810 + 2.228762210i\)
\(L(1)\) \(\approx\) \(1.196461047 + 0.6592294560i\)
\(L(1)\) \(\approx\) \(1.196461047 + 0.6592294560i\)

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

−21.52415827683957179623689671079, −20.71925447942360211003789091784, −20.14070271173987503190620891244, −19.2610069813026992380304570104, −18.7961662613985079402204772585, −17.65832384580732881298841722372, −16.97891919179595103576709684589, −16.01264284359210740851456392971, −15.05231846477880587601552086637, −14.22305089491121112498122482739, −13.71013920680552180006341135196, −12.78405738081055304724578514394, −12.14293922964949666068987666454, −10.66849223795124802528454527360, −10.28172664945795097654179500320, −9.083174566574143960172080792, −8.08174146923236115223995465453, −7.77605869816092889933442354531, −6.53896830371780428935490681749, −5.8014043023100791503109409471, −4.20004999552268975202379051391, −3.607975893302877523308788451271, −2.58204867061173302166492394753, −1.3611994440470848792626405374, −0.42134481896145305080555257678, 1.65505481805767116929737565904, 2.423003100294004604873275633144, 3.36117296601461950655829751250, 4.572788022242712626678944632489, 5.11968312433681392095443829216, 6.57875136282770770860188561664, 7.38520158756651284698645173408, 8.42833581429427614159540274038, 9.29278270995359054180895324309, 9.6296502844109400523244297085, 10.824369800296612516002794033107, 11.88780832805376205361966851350, 12.55251332416426536328641385441, 13.69547274645564004246317714225, 14.34624878208430235245343384996, 15.169905348608206259514875936120, 15.78667459868389767169638410796, 16.5485410828388972619923732247, 17.81910297035260688767373130155, 18.40458260069302661549504401722, 19.61779318667383235266794334844, 19.71455907212950007280737054880, 21.08644262796859199261931893776, 21.39103082900966791143589859062, 22.22801227467772730859461231391

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