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Properties

Label 1-800-800.731-r1-0-0
Degree $1$
Conductor $800$
Sign $0.744 + 0.667i$
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 − i·7-s + (−0.587 + 0.809i)9-s + (−0.987 − 0.156i)11-s + (−0.987 + 0.156i)13-s + (−0.309 − 0.951i)17-s + (−0.891 − 0.453i)19-s + (−0.891 + 0.453i)21-s + (−0.587 − 0.809i)23-s + (0.987 + 0.156i)27-s + (0.453 + 0.891i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.156 − 0.987i)37-s + (0.587 + 0.809i)39-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)3-s − i·7-s + (−0.587 + 0.809i)9-s + (−0.987 − 0.156i)11-s + (−0.987 + 0.156i)13-s + (−0.309 − 0.951i)17-s + (−0.891 − 0.453i)19-s + (−0.891 + 0.453i)21-s + (−0.587 − 0.809i)23-s + (0.987 + 0.156i)27-s + (0.453 + 0.891i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.156 − 0.987i)37-s + (0.587 + 0.809i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.744 + 0.667i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ 0.744 + 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09446482433 + 0.03611984500i\)
\(L(\frac12)\) \(\approx\) \(0.09446482433 + 0.03611984500i\)
\(L(1)\) \(\approx\) \(0.5669052557 - 0.3085304511i\)
\(L(1)\) \(\approx\) \(0.5669052557 - 0.3085304511i\)

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 \( 1 - iT \)
11 \( 1 + (-0.987 - 0.156i)T \)
13 \( 1 + (-0.987 + 0.156i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.891 - 0.453i)T \)
23 \( 1 + (-0.587 - 0.809i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
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 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.891 + 0.453i)T \)
59 \( 1 + (0.156 + 0.987i)T \)
61 \( 1 + (0.156 - 0.987i)T \)
67 \( 1 + (0.891 + 0.453i)T \)
71 \( 1 + (-0.951 - 0.309i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.891 - 0.453i)T \)
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

−21.92428472738511372452923445785, −21.313455378618477506605514253290, −20.631583437401330787415499896539, −19.53775299252889891598625559905, −18.807265975919353181165414683, −17.6639239952527967265672948860, −17.30156800821147008828408228573, −16.16888973553338628949263142757, −15.40773336729113619631990016535, −15.05022579080250518647138384327, −13.99875910360242485227929673513, −12.63880990690634368343689360843, −12.22637314622450736149356927718, −11.18259221335258553877183591878, −10.32197271612603942939468600990, −9.7244770539089831204335302500, −8.68307561137136465768633665477, −7.948786308129724187159529647994, −6.559348728934825810327203763056, −5.66695758188001742671619149096, −5.03290812590734130086788897909, −4.0492736380116036088775437480, −2.909052543372982878060015850265, −1.95792371569643903145955879354, −0.03661871467776540475939760042, 0.621005135125986619800976193730, 2.04779217838404424336821728861, 2.84604521596373992979040206011, 4.40140508155365080718233582225, 5.11359216381377005187645044013, 6.261420955397134916697833190410, 7.13556290762341912437440637531, 7.65882498848086024484567545250, 8.65014664185456752600889465898, 9.93710644633029271287560304022, 10.73866458885861362625167353964, 11.43141647740325401250764315059, 12.51780488054116488219762318993, 13.07830105561818572008683806110, 13.911801035392161754699345015811, 14.625806207170301494971284227463, 15.937792666785725675681784952541, 16.65063373581824179712784567566, 17.38220424851135865836118906383, 18.13712099712839680274249540045, 18.85320028701907723635074344278, 19.80495501628424556632156245438, 20.308602296031796963938974553853, 21.4421327517679376434409113744, 22.3234619446534407019428853026

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