Properties

Label 2-6e3-72.5-c2-0-3
Degree $2$
Conductor $216$
Sign $0.157 - 0.987i$
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  + (−1.99 − 0.171i)2-s + (3.94 + 0.681i)4-s + (−0.344 + 0.596i)5-s + (3.20 + 5.55i)7-s + (−7.73 − 2.03i)8-s + (0.788 − 1.13i)10-s + (2.32 + 4.03i)11-s + (−10.7 − 6.19i)13-s + (−5.43 − 11.6i)14-s + (15.0 + 5.37i)16-s + 26.4i·17-s − 11.2i·19-s + (−1.76 + 2.11i)20-s + (−3.94 − 8.42i)22-s + (1.52 + 0.882i)23-s + ⋯
L(s)  = 1  + (−0.996 − 0.0855i)2-s + (0.985 + 0.170i)4-s + (−0.0689 + 0.119i)5-s + (0.458 + 0.793i)7-s + (−0.967 − 0.254i)8-s + (0.0788 − 0.113i)10-s + (0.211 + 0.366i)11-s + (−0.825 − 0.476i)13-s + (−0.388 − 0.829i)14-s + (0.941 + 0.335i)16-s + 1.55i·17-s − 0.590i·19-s + (−0.0882 + 0.105i)20-s + (−0.179 − 0.383i)22-s + (0.0664 + 0.0383i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.157 - 0.987i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.157 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.650811 + 0.555287i\)
\(L(\frac12)\) \(\approx\) \(0.650811 + 0.555287i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 + 0.171i)T \)
3 \( 1 \)
good5 \( 1 + (0.344 - 0.596i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-3.20 - 5.55i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-2.32 - 4.03i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (10.7 + 6.19i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 26.4iT - 289T^{2} \)
19 \( 1 + 11.2iT - 361T^{2} \)
23 \( 1 + (-1.52 - 0.882i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-11.0 - 19.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (27.1 - 47.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 57.9iT - 1.36e3T^{2} \)
41 \( 1 + (-47.1 - 27.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-24.2 + 14.0i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-20.3 + 11.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 97.9T + 2.80e3T^{2} \)
59 \( 1 + (-38.4 + 66.6i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (0.493 - 0.284i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (58.9 + 34.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 59.1iT - 5.04e3T^{2} \)
73 \( 1 - 19.6T + 5.32e3T^{2} \)
79 \( 1 + (63.2 + 109. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-40.9 - 70.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 46.6iT - 7.92e3T^{2} \)
97 \( 1 + (-32.6 - 56.6i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.22037221053715978322389424794, −11.13591492073779733673092711740, −10.37483043690077883537198308790, −9.258163147957340270175182322984, −8.467924385538611051225634284914, −7.47349372415728685960366229610, −6.38482580773269435210331422866, −5.06827827171692157319907354863, −3.11435752681595414179504346631, −1.68374639394022978419498129652, 0.66228145878439984334251001939, 2.45510522907246910824496607031, 4.29929856949516600850374669123, 5.83089066928020086814245676879, 7.19254933780936346172263243351, 7.73930274163456032647184878150, 9.031885201812037183726267227325, 9.781606702042035971760745881313, 10.86622350836715319378572876796, 11.58541609527649025935121635124

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