↑

Properties

Label 1-3e5-243.223-r0-0-0
Degree $1$
Conductor $243$
Sign $0.852 - 0.522i$
Analytic cond. $1.12848$
Root an. cond. $1.12848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.565 − 0.824i)2-s + (−0.360 + 0.932i)4-s + (0.952 + 0.305i)5-s + (0.323 − 0.946i)7-s + (0.973 − 0.230i)8-s + (−0.286 − 0.957i)10-s + (0.466 + 0.884i)11-s + (0.0968 + 0.995i)13-s + (−0.963 + 0.268i)14-s + (−0.740 − 0.672i)16-s + (−0.835 − 0.549i)17-s + (0.893 + 0.448i)19-s + (−0.627 + 0.778i)20-s + (0.466 − 0.884i)22-s + (−0.981 − 0.192i)23-s + ⋯
L(s)  = 1  + (−0.565 − 0.824i)2-s + (−0.360 + 0.932i)4-s + (0.952 + 0.305i)5-s + (0.323 − 0.946i)7-s + (0.973 − 0.230i)8-s + (−0.286 − 0.957i)10-s + (0.466 + 0.884i)11-s + (0.0968 + 0.995i)13-s + (−0.963 + 0.268i)14-s + (−0.740 − 0.672i)16-s + (−0.835 − 0.549i)17-s + (0.893 + 0.448i)19-s + (−0.627 + 0.778i)20-s + (0.466 − 0.884i)22-s + (−0.981 − 0.192i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ 0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053956866 - 0.2970627150i\)
\(L(\frac12)\) \(\approx\) \(1.053956866 - 0.2970627150i\)
\(L(1)\) \(\approx\) \(0.9401576452 - 0.2491419595i\)
\(L(1)\) \(\approx\) \(0.9401576452 - 0.2491419595i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.565 - 0.824i)T \)
5 \( 1 + (0.952 + 0.305i)T \)
7 \( 1 + (0.323 - 0.946i)T \)
11 \( 1 + (0.466 + 0.884i)T \)
13 \( 1 + (0.0968 + 0.995i)T \)
17 \( 1 + (-0.835 - 0.549i)T \)
19 \( 1 + (0.893 + 0.448i)T \)
23 \( 1 + (-0.981 - 0.192i)T \)
29 \( 1 + (0.249 + 0.968i)T \)
31 \( 1 + (0.856 + 0.516i)T \)
37 \( 1 + (0.396 - 0.918i)T \)
41 \( 1 + (-0.431 - 0.902i)T \)
43 \( 1 + (0.925 + 0.378i)T \)
47 \( 1 + (0.856 - 0.516i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.999 + 0.0387i)T \)
61 \( 1 + (-0.360 - 0.932i)T \)
67 \( 1 + (0.249 - 0.968i)T \)
71 \( 1 + (-0.686 + 0.727i)T \)
73 \( 1 + (-0.286 + 0.957i)T \)
79 \( 1 + (0.996 - 0.0774i)T \)
83 \( 1 + (-0.431 + 0.902i)T \)
89 \( 1 + (-0.686 - 0.727i)T \)
97 \( 1 + (0.952 - 0.305i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.125868613793636334615008008616, −25.14013628136239109695464300391, −24.6159535711049904360122822326, −23.953371948513912144984095515979, −22.36635924364155544427673585817, −21.90199475419390536693208182213, −20.59013420544955212472748200238, −19.5459605028640002505221171139, −18.44573200943060386918357789107, −17.744153683418810508968801619656, −17.03348122482779281617471129002, −15.85105234978981336349066292088, −15.14650943371829182347702212676, −13.95398754956109108860744627864, −13.286754403517661605786987919640, −11.793525394336780556836492212356, −10.561364873509026944262337225917, −9.513032678453894002653973537050, −8.71237419841873288189950963863, −7.87407403776932891014668377228, −6.15481368999024367185029451675, −5.85346382835262218254300871147, −4.62191650566395358727129924927, −2.57952811521439220458787944226, −1.17992084251886046460904099078, 1.37632501464870201713536961895, 2.29783635839077713515947726699, 3.81477543359868333788440249526, 4.829677967467775767492152978470, 6.657577252173134508228935688138, 7.44951528281546691992938216344, 8.92484001279179030404582573216, 9.749472127874792345324094381719, 10.538496415983877029108229846608, 11.53748225769322914979789717474, 12.57801763720810355025294894113, 13.88253289320556755017299026436, 14.15817545778089402667925983948, 16.06141095194413575945451012906, 17.05817076434742382489526964907, 17.783660932295668814482222739587, 18.44466974386509377221034646628, 19.77903204698321195236671105573, 20.41649827470146909059638410126, 21.284748264457394015932896372488, 22.21377619139109901493264611421, 22.98151334011751802702494863364, 24.3768320720142882857378864811, 25.425933966683813642810713576720, 26.30931102063817816336166218362

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