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Properties

Label 2-6e3-8.5-c1-0-1
Degree $2$
Conductor $216$
Sign $-0.594 - 0.804i$
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  + (−1.38 + 0.297i)2-s + (1.82 − 0.822i)4-s + 3.36i·5-s − 2.64·7-s + (−2.27 + 1.68i)8-s + (−1 − 4.64i)10-s − 2.16i·11-s + 4.64i·13-s + (3.65 − 0.787i)14-s + (2.64 − 3i)16-s − 4.55·17-s + 6.29i·19-s + (2.76 + 6.12i)20-s + (0.645 + 3i)22-s − 0.979·23-s + ⋯
L(s)  = 1  + (−0.977 + 0.210i)2-s + (0.911 − 0.411i)4-s + 1.50i·5-s − 0.999·7-s + (−0.804 + 0.594i)8-s + (−0.316 − 1.46i)10-s − 0.654i·11-s + 1.28i·13-s + (0.977 − 0.210i)14-s + (0.661 − 0.750i)16-s − 1.10·17-s + 1.44i·19-s + (0.618 + 1.36i)20-s + (0.137 + 0.639i)22-s − 0.204·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.594 - 0.804i$
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: \((2,\ 216,\ (\ :1/2),\ -0.594 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251687 + 0.498719i\)
\(L(\frac12)\) \(\approx\) \(0.251687 + 0.498719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.297i)T \)
3 \( 1 \)
good5 \( 1 - 3.36iT - 5T^{2} \)
7 \( 1 + 2.64T + 7T^{2} \)
11 \( 1 + 2.16iT - 11T^{2} \)
13 \( 1 - 4.64iT - 13T^{2} \)
17 \( 1 + 4.55T + 17T^{2} \)
19 \( 1 - 6.29iT - 19T^{2} \)
23 \( 1 + 0.979T + 23T^{2} \)
29 \( 1 - 4.33iT - 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 1.35iT - 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 3.29iT - 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 4.33iT - 53T^{2} \)
59 \( 1 + 11.2iT - 59T^{2} \)
61 \( 1 + 1.93iT - 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 8.64T + 79T^{2} \)
83 \( 1 - 2.38iT - 83T^{2} \)
89 \( 1 - 4.55T + 89T^{2} \)
97 \( 1 - 2.29T + 97T^{2} \)
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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

−12.40002455767430245566626753008, −11.27671799479142045963156931542, −10.67050560659379782465759977785, −9.742981095586644460253535714996, −8.861584370310415841289139680015, −7.51936690481781020085459411077, −6.60090473603065494457554458321, −6.08514839981838820456301681401, −3.62267211503731907806820742654, −2.33918178296509285885561530142, 0.61650717592624954793526747740, 2.61366175096477685380464225980, 4.39341080395886689399795682458, 5.86835198961783838729791337594, 7.12212909795651284788086666652, 8.255981905311484117140600037972, 9.133209593933303672628579104867, 9.744128986774655844102150634607, 10.85823582643567279930031611996, 12.04197864637880995076657149922

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