↑

Properties

Label 1-6e3-216.13-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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 − 0.984i)5-s + (0.766 + 0.642i)7-s + (−0.173 + 0.984i)11-s + (0.939 + 0.342i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.939 − 0.342i)29-s + (0.766 − 0.642i)31-s + (0.5 − 0.866i)35-s + (0.5 + 0.866i)37-s + (−0.939 − 0.342i)41-s + (−0.173 + 0.984i)43-s + (0.766 + 0.642i)47-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)5-s + (0.766 + 0.642i)7-s + (−0.173 + 0.984i)11-s + (0.939 + 0.342i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.939 − 0.342i)29-s + (0.766 − 0.642i)31-s + (0.5 − 0.866i)35-s + (0.5 + 0.866i)37-s + (−0.939 − 0.342i)41-s + (−0.173 + 0.984i)43-s + (0.766 + 0.642i)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: $\chi_{216} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (0:\ ),\ 0.957 - 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.257866916 - 0.1842503730i\)
\(L(\frac12)\) \(\approx\) \(1.257866916 - 0.1842503730i\)
\(L(1)\) \(\approx\) \(1.135363809 - 0.09577605295i\)
\(L(1)\) \(\approx\) \(1.135363809 - 0.09577605295i\)

Euler product

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

−26.95188956029754060523053423184, −25.80685209660666627360951614566, −24.79578001885283717632607509163, −23.589177606125501275310849147469, −23.13809898026275230811989582707, −21.855998048196325018763399448974, −21.143064313891686317183484310679, −20.01230855770255415142473746397, −18.9942067186433439989229575748, −18.15190780081587066203873968946, −17.27530335273719602714179881277, −16.053586712022488101087773817742, −15.09421142517076935948597231673, −14.07052058418697392133617903678, −13.4162630020725314433630169463, −11.831813300231638549779951996856, −10.85196116421502373175456754375, −10.39082709952588641905667309733, −8.63679167029275999749472503554, −7.799062280184697706659853336845, −6.64266898929762676301546705797, −5.56570127435706559664697645682, −3.99633353539115871576593529286, −3.075750905591395117690663826881, −1.36992684820515898410561699280, 1.24994491370326922680253222886, 2.61755500908425245293553670510, 4.50203358369391784250635296555, 5.002317028958204641224530108689, 6.48863941473506986484563432000, 7.85560748970738410821159479808, 8.77022763049119369400888738970, 9.6116306914776663179135668210, 11.19297943365634251775486933950, 11.930254069834562469692060382908, 12.97559519831054976245828938300, 13.93976983923118826841834393541, 15.317149719914611086127217145263, 15.856792212076318351062470700798, 17.14587123432016922549978538219, 17.95040889521677191077228052588, 18.91794269145315822670708007075, 20.33015109574250703128468501411, 20.67940107446380854743517370848, 21.74771674130143027385851456529, 22.94590534617130625927546544517, 23.837587186255422669056248375581, 24.72539320865827943825883182628, 25.38212458959143275535756928205, 26.64544518685096762244152203042

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