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Properties

Label 2-6e3-9.2-c2-0-1
Degree $2$
Conductor $216$
Sign $0.449 - 0.893i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 + 1.04i)5-s + (−0.781 + 1.35i)7-s + (10.8 + 6.25i)11-s + (11.0 + 19.1i)13-s + 12.6i·17-s − 21.7·19-s + (28.7 − 16.6i)23-s + (−10.3 + 17.8i)25-s + (25.7 + 14.8i)29-s + (6.91 + 11.9i)31-s − 3.26i·35-s − 8.26·37-s + (−43.8 + 25.3i)41-s + (35.5 − 61.5i)43-s + (−57.2 − 33.0i)47-s + ⋯
L(s)  = 1  + (−0.361 + 0.208i)5-s + (−0.111 + 0.193i)7-s + (0.984 + 0.568i)11-s + (0.849 + 1.47i)13-s + 0.747i·17-s − 1.14·19-s + (1.25 − 0.722i)23-s + (−0.412 + 0.714i)25-s + (0.888 + 0.513i)29-s + (0.223 + 0.386i)31-s − 0.0932i·35-s − 0.223·37-s + (−1.06 + 0.617i)41-s + (0.826 − 1.43i)43-s + (−1.21 − 0.703i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.449 - 0.893i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.449 - 0.893i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19931 + 0.739302i\)
\(L(\frac12)\) \(\approx\) \(1.19931 + 0.739302i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (1.80 - 1.04i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (0.781 - 1.35i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-10.8 - 6.25i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-11.0 - 19.1i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 12.6iT - 289T^{2} \)
19 \( 1 + 21.7T + 361T^{2} \)
23 \( 1 + (-28.7 + 16.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-25.7 - 14.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-6.91 - 11.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 8.26T + 1.36e3T^{2} \)
41 \( 1 + (43.8 - 25.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-35.5 + 61.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (57.2 + 33.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 6.04iT - 2.80e3T^{2} \)
59 \( 1 + (-8.01 + 4.62i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-51.9 + 89.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (19.8 + 34.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 18.3iT - 5.04e3T^{2} \)
73 \( 1 + 68.5T + 5.32e3T^{2} \)
79 \( 1 + (-13.3 + 23.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (21.0 + 12.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 111. iT - 7.92e3T^{2} \)
97 \( 1 + (2.51 - 4.35i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.17147428044101618750288274786, −11.35266706694385760331636597660, −10.43632611523014254722694165541, −9.122225951161095215475969234120, −8.521682267951173862909690823312, −6.93588578560565444089090923880, −6.38099039840723624812860544896, −4.63843602866098716391908460851, −3.61770792692957709534675385333, −1.73986901727507146758740957247, 0.858767024485041830863964339343, 3.08080513970757130848849454877, 4.28124105892902277117401275342, 5.71183610099856659424115982372, 6.76586235286894060762357169323, 8.072356494693650621305360227147, 8.808394460739566575280402942495, 10.03177275733881806492041219459, 11.03996082756367380113474079388, 11.83639063472794072543693257204

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