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Properties

Label 1-800-800.197-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  + (−0.453 + 0.891i)3-s − 7-s + (−0.587 − 0.809i)9-s + (0.156 + 0.987i)11-s + (0.987 + 0.156i)13-s + (−0.951 − 0.309i)17-s + (0.453 + 0.891i)19-s + (0.453 − 0.891i)21-s + (0.809 + 0.587i)23-s + (0.987 − 0.156i)27-s + (0.891 + 0.453i)29-s + (0.309 − 0.951i)31-s + (−0.951 − 0.309i)33-s + (0.156 − 0.987i)37-s + (−0.587 + 0.809i)39-s + ⋯
L(s)  = 1  + (−0.453 + 0.891i)3-s − 7-s + (−0.587 − 0.809i)9-s + (0.156 + 0.987i)11-s + (0.987 + 0.156i)13-s + (−0.951 − 0.309i)17-s + (0.453 + 0.891i)19-s + (0.453 − 0.891i)21-s + (0.809 + 0.587i)23-s + (0.987 − 0.156i)27-s + (0.891 + 0.453i)29-s + (0.309 − 0.951i)31-s + (−0.951 − 0.309i)33-s + (0.156 − 0.987i)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: $\chi_{800} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ -0.943 + 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1840343227 + 1.079480969i\)
\(L(\frac12)\) \(\approx\) \(0.1840343227 + 1.079480969i\)
\(L(1)\) \(\approx\) \(0.7495243194 + 0.3635419867i\)
\(L(1)\) \(\approx\) \(0.7495243194 + 0.3635419867i\)

Euler product

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

−22.09628691064067647119082139060, −20.883629809232269968762207887905, −19.82349153106262629660407865921, −19.24292841096289966121358443410, −18.5895538099452227179996276778, −17.72101320234488532042517277094, −16.957559949919831599029213154188, −16.07237970341973414619286578261, −15.49790549460257752263821330043, −14.00258821916332170955413900108, −13.44758950799729737787191383064, −12.8513924052524306162000673763, −11.87473070066493460765780342613, −11.056564050783227039167306193730, −10.37340206329049758183454993964, −8.91598413355180105542214913203, −8.49520943535729055684079914010, −7.16583174246885595815075521262, −6.459160491016899813065473503251, −5.90180599074187893615482158991, −4.7195716612206898152453006378, −3.357799797448038233559139853364, −2.57519468458762757198921759752, −1.125775132992348314348162442483, −0.33343284535186187910850553836, 1.05197540483533826907102879080, 2.622955165477715579623405925, 3.67341932030614668155204060097, 4.368875820354384054286634006501, 5.465479644505842683937319905863, 6.333412768535449256196436763540, 7.08363088838881078856306190074, 8.48125376001809691045382202726, 9.43323393121758111816640785702, 9.872722319778744256864907864192, 10.88101042971984937787836916580, 11.63289218266029201993880667345, 12.58060575172690930164648155170, 13.38471796870425175962236743718, 14.467269048600464746275690004892, 15.37118037113841322382909644611, 15.97485309286318526507454583391, 16.59568794998464484313980856036, 17.57966468780330608285221615652, 18.23240902935058649806479734680, 19.34496074335724725006427675807, 20.18917899040770684213312873169, 20.81633486949178848895442394384, 21.68874912557905184282145069515, 22.57804033367811067230492745317

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