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Properties

Label 1-801-801.481-r0-0-0
Degree $1$
Conductor $801$
Sign $-0.398 + 0.916i$
Analytic cond. $3.71982$
Root an. cond. $3.71982$
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  + (−0.995 − 0.0950i)2-s + (0.981 + 0.189i)4-s + (−0.235 − 0.971i)5-s + (−0.690 − 0.723i)7-s + (−0.959 − 0.281i)8-s + (0.142 + 0.989i)10-s + (0.235 − 0.971i)11-s + (−0.945 − 0.327i)13-s + (0.618 + 0.786i)14-s + (0.928 + 0.371i)16-s + (−0.415 + 0.909i)17-s + (−0.989 − 0.142i)19-s + (−0.0475 − 0.998i)20-s + (−0.327 + 0.945i)22-s + (−0.371 − 0.928i)23-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)2-s + (0.981 + 0.189i)4-s + (−0.235 − 0.971i)5-s + (−0.690 − 0.723i)7-s + (−0.959 − 0.281i)8-s + (0.142 + 0.989i)10-s + (0.235 − 0.971i)11-s + (−0.945 − 0.327i)13-s + (0.618 + 0.786i)14-s + (0.928 + 0.371i)16-s + (−0.415 + 0.909i)17-s + (−0.989 − 0.142i)19-s + (−0.0475 − 0.998i)20-s + (−0.327 + 0.945i)22-s + (−0.371 − 0.928i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 801 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 801 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(801\)    =    \(3^{2} \cdot 89\)
Sign: $-0.398 + 0.916i$
Analytic conductor: \(3.71982\)
Root analytic conductor: \(3.71982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{801} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 801,\ (0:\ ),\ -0.398 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03214985387 - 0.04904740498i\)
\(L(\frac12)\) \(\approx\) \(-0.03214985387 - 0.04904740498i\)
\(L(1)\) \(\approx\) \(0.4413468711 - 0.1774119312i\)
\(L(1)\) \(\approx\) \(0.4413468711 - 0.1774119312i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.995 - 0.0950i)T \)
5 \( 1 + (-0.235 - 0.971i)T \)
7 \( 1 + (-0.690 - 0.723i)T \)
11 \( 1 + (0.235 - 0.971i)T \)
13 \( 1 + (-0.945 - 0.327i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (-0.989 - 0.142i)T \)
23 \( 1 + (-0.371 - 0.928i)T \)
29 \( 1 + (0.690 + 0.723i)T \)
31 \( 1 + (-0.618 - 0.786i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.189 - 0.981i)T \)
43 \( 1 + (-0.971 - 0.235i)T \)
47 \( 1 + (0.327 + 0.945i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.189 + 0.981i)T \)
61 \( 1 + (0.998 - 0.0475i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (-0.928 - 0.371i)T \)
83 \( 1 + (0.814 - 0.580i)T \)
97 \( 1 + (0.723 - 0.690i)T \)
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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

−22.88738865578987793291088764718, −21.870287342413948036209844615941, −21.31423312115350478686028066607, −19.80776750592618351911970348132, −19.69411691885696472124277815253, −18.736018391935973967204432974600, −18.05617723209829978856688114492, −17.40400391906332113657770920536, −16.366860285945019683061332189745, −15.53822095411678938346709720620, −14.99623120116203230483353399374, −14.20568753127557995197653731068, −12.75183111049640097948903168471, −11.90783789817870207987355386597, −11.32097684005221067298370722689, −10.08529671659321149634974467183, −9.73976727432282926714545395268, −8.80046946029999343562642722623, −7.68294080193777073856323141299, −6.90147484194123289175299378403, −6.403101499054193176537030316311, −5.15784969650275839680501047383, −3.69885447913543706143179713014, −2.535439480319836975577093600990, −2.038670323456838176236265348305, 0.04106342181463723972772267005, 1.0427111556089374477348033359, 2.32045299915646804864900385892, 3.51115987652754889645565590245, 4.452527333763926783212161795481, 5.8549124320437706420583689118, 6.645941915843389887378975623312, 7.65984783808209403257359144174, 8.521369216760966816933320068622, 9.068287782926469906382834708224, 10.18538776658499453520933337575, 10.69697675839973588028644226408, 11.8349342074358790992184277965, 12.61690491143154144412574444367, 13.27995004134571056599511506783, 14.58020002267057705619815879466, 15.54513451086232977522116679573, 16.38164767218016489351627683658, 16.936153055037834527724444258245, 17.38993626561189886234316162474, 18.69375771196825630082156290609, 19.45215468479355788509720566938, 19.896770465839197064649837269311, 20.60864300647206526345054540556, 21.58358209147128191635566756407

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