Properties

Label 2-6e3-8.5-c1-0-4
Degree $2$
Conductor $216$
Sign $1$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.00·4-s + 4.41i·5-s + 3.24·7-s + 2.82i·8-s + 6.24·10-s + 0.171i·11-s − 4.58i·14-s + 4.00·16-s − 8.82i·20-s + 0.242·22-s − 14.4·25-s − 6.48·28-s − 2.82i·29-s + 9.24·31-s − 5.65i·32-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + 1.97i·5-s + 1.22·7-s + 1.00i·8-s + 1.97·10-s + 0.0517i·11-s − 1.22i·14-s + 1.00·16-s − 1.97i·20-s + 0.0517·22-s − 2.89·25-s − 1.22·28-s − 0.525i·29-s + 1.66·31-s − 1.00i·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18774\)
\(L(\frac12)\) \(\approx\) \(1.18774\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
good5 \( 1 - 4.41iT - 5T^{2} \)
7 \( 1 - 3.24T + 7T^{2} \)
11 \( 1 - 0.171iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 - 9.24T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 4.07iT - 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 17.8iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.80774928306062185451101436052, −11.37522714816070148754386847285, −10.52979752939071557861302780712, −9.864665624649668808362080951209, −8.353079900643625565664156056767, −7.43320751867388141567271288795, −6.08030678483194142331386839723, −4.56654284062922243129795878754, −3.23425853026846556284851884959, −2.09347155128834804444612063635, 1.20357386349153630305809421009, 4.32242328711747486569045006635, 4.93396269287804346789627372966, 5.87698713383913919953825076736, 7.54840850903100823415266873109, 8.406152479796825356247832892286, 8.890519749902944527139694254697, 10.03784094777707466947767850714, 11.65210918947207506448662675421, 12.48815097749150866335251596175

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