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Properties

Label 1-555-555.77-r0-0-0
Degree $1$
Conductor $555$
Sign $0.766 + 0.642i$
Analytic cond. $2.57740$
Root an. cond. $2.57740$
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  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.642 + 0.766i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.5 − 0.866i)14-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + (0.173 − 0.984i)19-s + (−0.642 − 0.766i)22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)26-s + (−0.342 + 0.939i)28-s + (0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.642 + 0.766i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.5 − 0.866i)14-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + (0.173 − 0.984i)19-s + (−0.642 − 0.766i)22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)26-s + (−0.342 + 0.939i)28-s + (0.5 + 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(2.57740\)
Root analytic conductor: \(2.57740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (0:\ ),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7559968221 + 0.2750147744i\)
\(L(\frac12)\) \(\approx\) \(0.7559968221 + 0.2750147744i\)
\(L(1)\) \(\approx\) \(0.6911530795 + 0.1140054702i\)
\(L(1)\) \(\approx\) \(0.6911530795 + 0.1140054702i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.984 + 0.173i)T \)
7 \( 1 + (-0.642 + 0.766i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.642 + 0.766i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.342 - 0.939i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−23.39549078070430282053359627915, −22.25972034367069945197010761580, −21.374987934243239997212335268977, −20.65147288709041048540510245250, −19.55613830149832615439110903029, −19.20404448378597427977102637420, −18.395158330596058471698627471011, −17.04358704085966738750559722974, −16.79971613782939714609919481254, −16.03797395707785294382571509083, −14.83012727019959242001753774456, −13.939420112624213502274371403713, −12.79576578803700810530727810716, −11.9454465917085845405684309688, −10.97477606563327375844955351006, −10.23796690748287939491231816700, −9.36688546435606095381939275850, −8.537935467535433915965889358802, −7.52000128261891318842462976897, −6.649609239057718203283256445670, −5.87058695681720695691369977018, −4.06280970493492779317049705788, −3.308283382271801709953841069662, −1.94958477688021075642945379575, −0.7623959685060520729947329258, 0.97262580559913868582351735474, 2.421991208074431308194355010319, 3.1577541447577426740572999655, 4.963712035152528560372982130714, 5.84893797710791947222787398319, 7.001027069175366037511074184098, 7.531956597172606709811933746998, 8.95024851224585245731454221996, 9.336393243799507729152655547, 10.261573106793705827790735408661, 11.292448957632119758024237548978, 12.19762134819244698569108148, 12.935764205583824834941982335850, 14.41407085125146788390907667645, 15.24342150910793047867885904379, 15.84182075826601475534766150969, 16.76729001903075571046378711587, 17.74117340340819978946992202736, 18.2138718433308697485026373335, 19.29676901301087135160624129925, 19.86863381788844185726348869705, 20.64892649380550957932418622017, 21.75668921970707398162143619794, 22.59925600282085640513819980706, 23.48907835911237461537796117651

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