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Properties

Label 2-6e3-24.11-c1-0-9
Degree $2$
Conductor $216$
Sign $0.941 - 0.335i$
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  + (−0.563 + 1.29i)2-s + (−1.36 − 1.46i)4-s + 3.07·5-s − 3.99i·7-s + (2.66 − 0.949i)8-s + (−1.73 + 3.99i)10-s − 1.89i·11-s − 1.06i·13-s + (5.17 + 2.24i)14-s + (−0.267 + 3.99i)16-s + 7.08i·17-s + 3.73·19-s + (−4.20 − 4.49i)20-s + (2.46 + 1.06i)22-s − 0.824·23-s + ⋯
L(s)  = 1  + (−0.398 + 0.917i)2-s + (−0.683 − 0.730i)4-s + 1.37·5-s − 1.50i·7-s + (0.941 − 0.335i)8-s + (−0.547 + 1.26i)10-s − 0.572i·11-s − 0.296i·13-s + (1.38 + 0.600i)14-s + (−0.0669 + 0.997i)16-s + 1.71i·17-s + 0.856·19-s + (−0.939 − 1.00i)20-s + (0.525 + 0.227i)22-s − 0.171·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.941 - 0.335i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.941 - 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13427 + 0.196118i\)
\(L(\frac12)\) \(\approx\) \(1.13427 + 0.196118i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.563 - 1.29i)T \)
3 \( 1 \)
good5 \( 1 - 3.07T + 5T^{2} \)
7 \( 1 + 3.99iT - 7T^{2} \)
11 \( 1 + 1.89iT - 11T^{2} \)
13 \( 1 + 1.06iT - 13T^{2} \)
17 \( 1 - 7.08iT - 17T^{2} \)
19 \( 1 - 3.73T + 19T^{2} \)
23 \( 1 + 0.824T + 23T^{2} \)
29 \( 1 + 4.50T + 29T^{2} \)
31 \( 1 - 5.84iT - 31T^{2} \)
37 \( 1 + 6.91iT - 37T^{2} \)
41 \( 1 - 3.79iT - 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 6.97T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + 1.89iT - 59T^{2} \)
61 \( 1 - 1.06iT - 61T^{2} \)
67 \( 1 + 9.19T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 5.92T + 73T^{2} \)
79 \( 1 - 6.12iT - 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 - 13.6iT - 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−12.87688502542768942921915967460, −10.84387878637064537260976552882, −10.29943491374386828063810948617, −9.469577812174528614624454106509, −8.319211467268061826568309833953, −7.25985797935928599322466925954, −6.23684492566215660514928319023, −5.39258725798301041641202357656, −3.87867850793446486134299053219, −1.37559202972146124650397991372, 1.93353091808654697566927918225, 2.84439020944183745833256534656, 4.89625764763831413271411378840, 5.81186437060455358134724394832, 7.36576342716726939924196078475, 8.849158978062261741204115900602, 9.464813137580920002944114367243, 10.02362856240276656243010246691, 11.51951536934245202117454532446, 12.01630261412972280204928295845

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