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Properties

Label 1-6e3-216.85-r0-0-0
Degree $1$
Conductor $216$
Sign $0.957 - 0.286i$
Analytic cond. $1.00309$
Root an. cond. $1.00309$
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  + (0.939 + 0.342i)5-s + (0.173 − 0.984i)7-s + (0.939 − 0.342i)11-s + (−0.766 + 0.642i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.766 − 0.642i)29-s + (0.173 + 0.984i)31-s + (0.5 − 0.866i)35-s + (0.5 + 0.866i)37-s + (0.766 − 0.642i)41-s + (0.939 − 0.342i)43-s + (0.173 − 0.984i)47-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)5-s + (0.173 − 0.984i)7-s + (0.939 − 0.342i)11-s + (−0.766 + 0.642i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.766 − 0.642i)29-s + (0.173 + 0.984i)31-s + (0.5 − 0.866i)35-s + (0.5 + 0.866i)37-s + (0.766 − 0.642i)41-s + (0.939 − 0.342i)43-s + (0.173 − 0.984i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.957 - 0.286i$
Analytic conductor: \(1.00309\)
Root analytic conductor: \(1.00309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (0:\ ),\ 0.957 - 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.399782117 - 0.2050378890i\)
\(L(\frac12)\) \(\approx\) \(1.399782117 - 0.2050378890i\)
\(L(1)\) \(\approx\) \(1.243001482 - 0.08736687500i\)
\(L(1)\) \(\approx\) \(1.243001482 - 0.08736687500i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.939 - 0.342i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 - T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−26.60729313268245805189110355296, −25.46433842173186562754451378691, −24.79308695087611625894009759326, −24.23238224091573616713917381850, −22.488172484918651426343140050674, −22.12051527050117468418385715828, −21.03470156228850134645251589810, −20.16495622047404023307770869066, −19.03703844657405366876074236118, −17.95751850023353321029323613816, −17.26373242493488472741794211714, −16.26474763152934807590452674710, −14.876216295909615252032628833565, −14.38089994774978379094177508725, −12.84047698040468843350836466144, −12.36626887657984883031278760219, −11.004766485573721429593863376658, −9.73131452105202654063281609731, −9.06419244447530307690609588165, −7.88103067519264134548487104494, −6.34038323552772057135347923293, −5.58015860861037134997756721518, −4.36601336058216746288952252502, −2.65285632240493388150360608080, −1.58795394627908221994923863083, 1.29686704576777399592908182690, 2.70370535921491638341958050310, 4.12971599740316084684361346365, 5.32782687501564880469006185078, 6.706524971798992612064222601691, 7.32719440594959220948252596804, 9.08422536500417384458031134524, 9.73675015107625849925314235602, 10.94801131978884679273194444005, 11.80037012035717577494008059620, 13.412448958296992132001097943615, 13.89037506256084456808415082054, 14.81203063142040867012835937391, 16.23996243767016060297623517725, 17.23617225155170044059484437937, 17.731930035614560283279618189161, 19.081687588645729354914711693201, 19.95128928812473766976899446089, 20.97209745770959770159840178573, 21.96056661275046697123439243728, 22.60532275786439918602937568459, 23.91152249213113814258493880907, 24.655950523251800215249511706705, 25.65425340868034249142142150638, 26.64552208532618089158546556162

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