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Properties

Label 1-800-800.491-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  + (0.156 − 0.987i)3-s − i·7-s + (−0.951 − 0.309i)9-s + (0.453 − 0.891i)11-s + (0.453 + 0.891i)13-s + (0.809 + 0.587i)17-s + (−0.987 + 0.156i)19-s + (−0.987 − 0.156i)21-s + (−0.951 + 0.309i)23-s + (−0.453 + 0.891i)27-s + (−0.156 + 0.987i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.891 + 0.453i)37-s + (0.951 − 0.309i)39-s + ⋯
L(s)  = 1  + (0.156 − 0.987i)3-s − i·7-s + (−0.951 − 0.309i)9-s + (0.453 − 0.891i)11-s + (0.453 + 0.891i)13-s + (0.809 + 0.587i)17-s + (−0.987 + 0.156i)19-s + (−0.987 − 0.156i)21-s + (−0.951 + 0.309i)23-s + (−0.453 + 0.891i)27-s + (−0.156 + 0.987i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.891 + 0.453i)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: $\chi_{800} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ 0.389 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5583814223 + 0.3699350790i\)
\(L(\frac12)\) \(\approx\) \(0.5583814223 + 0.3699350790i\)
\(L(1)\) \(\approx\) \(0.8939142336 - 0.3147291085i\)
\(L(1)\) \(\approx\) \(0.8939142336 - 0.3147291085i\)

Euler product

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

−21.88916973630609881337702362236, −21.08207887546743792036218311479, −20.49932572293883188596155244978, −19.64841273661273252169940071607, −18.74416755922245355627512567419, −17.80430113036372625445542766992, −17.06073775010802249661443934436, −16.08921624440116326187366814058, −15.34851968247786876655028601824, −14.88503012733247269283151719587, −13.985354304025409535628710100111, −12.83799630876080907964024542402, −11.96501452896985277935885798185, −11.249657381586425629157942272301, −10.0089270744509869009154625422, −9.73475044706201118240740216319, −8.54007476381995119067135484848, −8.04259570412571877027926171646, −6.54092599021145485907361898706, −5.65521529057614382559036127925, −4.84174997756452526388275641195, −3.88112171390549112829018385131, −2.881848461328577320426436077672, −1.94230315929908253933146063203, −0.14713596968056293859157521849, 1.13490116067599738231565903261, 1.80679958590016155669745034591, 3.31830340699795792023345337767, 3.9487808619695561371013975535, 5.390640830054167791982989678129, 6.524646029753326983161248965893, 6.8468186524075038540530156984, 8.18700654560583911606033471843, 8.504790210910413182233854867480, 9.82039685256493438384037351288, 10.81346200847739649299815602661, 11.60179501035818673177218444692, 12.449214605622757615183673840026, 13.34930457647069118847192821877, 14.09281217658582424981817988786, 14.459062934308871074534879684268, 15.90292405105705900543398494586, 16.86275903864233641163296338132, 17.22987184553308275957257102358, 18.357340441542165668444873623958, 19.12300941141009581178465766904, 19.57460981926467040911377583963, 20.53323608245289767746464578304, 21.3289160535101721025156166229, 22.2900687785041351156446084948

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