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Properties

Label 1-800-800.717-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: $\chi_{800} (717, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ 0.943 + 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.794451862 + 0.4764095624i\)
\(L(\frac12)\) \(\approx\) \(2.794451862 + 0.4764095624i\)
\(L(1)\) \(\approx\) \(1.392565215 + 0.2771495198i\)
\(L(1)\) \(\approx\) \(1.392565215 + 0.2771495198i\)

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

−22.09363301512185483285639310242, −20.919004295313945985881722607831, −20.44386030402384159015829061778, −19.63743364455951999699599265099, −18.77953136696208242039062685153, −17.998126033023667612609591910217, −17.43620724155224707656229275688, −16.58156080822914063427414103867, −15.20926316537500082728859537126, −14.51441080035064927495856037543, −14.14772382396264525247096686513, −12.93494864148520913721005164433, −12.11414380750437009467566734294, −11.76792163564605562777014457391, −10.30293226041278084312971797466, −9.62511779523914231578296319947, −8.353039786675309977730872611097, −7.82878665601781958350106418413, −7.105873230244028794628109547834, −6.01456887311435436326972822525, −4.99826381587527941787687438466, −3.969877937793995468426453951308, −2.67143315826890265374776715714, −1.851323035802351486068965505274, −0.92323922267787021865480625925, 0.73188988550067790681830822612, 2.1837104299703785052781674937, 3.108960514035620870872363808606, 4.12372620587704906965824180696, 5.04749619677273798709694582786, 5.67162216903369304677636873312, 7.2047773911212158750162091503, 8.04650972070541421387136133511, 8.78990660571826056833833730227, 9.69007997151825119935543086628, 10.475989456704507494857677480374, 11.42017153645521791363021854136, 11.98134460305721016115091004942, 13.43182176679725313532714116560, 14.33832602474433986647359370084, 14.50155088870721537494695742917, 15.779641673848413527540079598962, 16.2403962904245644348635187432, 17.30628672767710289408923061917, 17.89951198733550011914646782711, 19.25569956580151311403408900045, 19.63325866483431270899376536066, 20.6866809099543041951928512453, 21.4067779647024604172468453040, 21.77851043996586589708417630438

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