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Properties

Label 1-800-800.397-r1-0-0
Degree $1$
Conductor $800$
Sign $-0.667 - 0.744i$
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  + (−0.891 − 0.453i)3-s + 7-s + (0.587 + 0.809i)9-s + (0.987 − 0.156i)11-s + (−0.156 + 0.987i)13-s + (−0.951 − 0.309i)17-s + (−0.891 + 0.453i)19-s + (−0.891 − 0.453i)21-s + (−0.809 − 0.587i)23-s + (−0.156 − 0.987i)27-s + (−0.453 + 0.891i)29-s + (0.309 − 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.987 − 0.156i)37-s + (0.587 − 0.809i)39-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)3-s + 7-s + (0.587 + 0.809i)9-s + (0.987 − 0.156i)11-s + (−0.156 + 0.987i)13-s + (−0.951 − 0.309i)17-s + (−0.891 + 0.453i)19-s + (−0.891 − 0.453i)21-s + (−0.809 − 0.587i)23-s + (−0.156 − 0.987i)27-s + (−0.453 + 0.891i)29-s + (0.309 − 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.987 − 0.156i)37-s + (0.587 − 0.809i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.667 - 0.744i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ -0.667 - 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3169423138 - 0.7093636459i\)
\(L(\frac12)\) \(\approx\) \(0.3169423138 - 0.7093636459i\)
\(L(1)\) \(\approx\) \(0.7992405240 - 0.1329053770i\)
\(L(1)\) \(\approx\) \(0.7992405240 - 0.1329053770i\)

Euler product

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

−22.21901405998198176331603115639, −21.768441076602038323737547490878, −20.84262611534672821270653928154, −20.09749961587591716271236988694, −19.17113863868965368657294626598, −17.95716061788508412955599246358, −17.452733306034003251499552165799, −17.05453908377799361448486865038, −15.75104859684942818385315625704, −15.23374888037925520890520583847, −14.42028802002792405229615548649, −13.32677742319177271666996603985, −12.30558562160523621943841660779, −11.62943310379709983816534535997, −10.86749643370577369969342273842, −10.174293779106103394024647636637, −9.10390993094005844809964192980, −8.24562182002107089682489079260, −7.09878509721351603942485661291, −6.232354557972790104919185150, −5.32209498163852844875030325260, −4.470049659925154718009335432205, −3.73749001827598379616512649351, −2.13566929833875128676135577459, −1.046760355461448546303739997054, 0.21434110773433658945836592033, 1.557844615318847433103521115292, 2.15070148798835946909749745287, 4.08407402439183469300839695983, 4.594051997720414986462593049141, 5.75057212259897274052758644878, 6.571717111826425364950410943067, 7.30449438042398896728912174785, 8.39110357234184098357608904099, 9.213072585004776151585282859229, 10.500862511149999859147343963108, 11.17932347316472307954820109453, 11.87932726455192716619826063063, 12.51703767030275130555730382827, 13.74084914603540532826809835349, 14.28726110403277743579124578333, 15.30965675562804298715072401013, 16.39862202160641271058342084846, 17.0382133913163930856572659727, 17.64682365015355637076750900424, 18.53829775916000727648708134287, 19.153303689872419490308517936836, 20.1760438710376392938435019572, 21.09902921233442598275374662465, 21.97558948631825330738783553731

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