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Properties

Label 1-800-800.203-r0-0-0
Degree $1$
Conductor $800$
Sign $0.943 - 0.331i$
Analytic cond. $3.71518$
Root an. cond. $3.71518$
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

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

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(3.71518\)
Root analytic conductor: \(3.71518\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (0:\ ),\ 0.943 - 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.564890101 - 0.2667888534i\)
\(L(\frac12)\) \(\approx\) \(1.564890101 - 0.2667888534i\)
\(L(1)\) \(\approx\) \(1.176460567 - 0.2135914542i\)
\(L(1)\) \(\approx\) \(1.176460567 - 0.2135914542i\)

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

−22.30434955430763241589215907108, −21.51078585192475316540638266374, −20.66307251317736102418968395648, −20.1001723832411680319312914670, −18.97915577322678678994388953062, −18.71567588165450980322825887111, −17.19562182425847401877177340960, −16.44207868204857959452981831044, −15.9357959045383751320487216409, −15.155443175079268114678532884935, −14.139509856873528641640699758993, −13.492526911384214054537941087404, −12.667172833509268223642603209486, −11.298354234991355832766630602334, −10.8313977352943789232451513351, −9.71616739986069479311449095129, −9.14696360302764131953794020734, −8.35233706765050951465420901982, −7.26137798308098764298792414149, −6.06682217407497231204502785135, −5.38642717806264489963518580811, −4.11343453148609494065001774351, −3.3037168997172482154742390367, −2.70440579497188739389730290229, −0.8721029348536107078872622582, 1.07331531458267205083056637467, 2.03139654512712820433984274566, 3.26979189687397718020628051038, 3.85143949480079200101432890811, 5.52342712399884843335194541249, 6.26287680965498834342314819667, 7.1996790707245220572753415145, 7.81508977674867091796939313302, 9.01361792743997393923233207948, 9.56723627229344401071864443504, 10.64486932767953015700186370356, 11.837078480779564063952854177223, 12.56181309187655294637717053, 13.12195145896467848302784160236, 14.03717207367121668613304409234, 14.7916130613090582978804606681, 15.73092369447890599219878728181, 16.625160239263284931410401975036, 17.51303595272677106731842062584, 18.41113921672592863460655025413, 18.9956370584325379705339027001, 19.71884343902956687539022183646, 20.53195762769872787886017206744, 21.201523404615499026253427444622, 22.517118258519888819447232274281

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