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Properties

Label 2-6e3-72.43-c2-0-8
Degree $2$
Conductor $216$
Sign $0.587 + 0.809i$
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  + (−1.89 − 0.629i)2-s + (3.20 + 2.39i)4-s + (−5.84 + 3.37i)5-s + (−3.50 − 2.02i)7-s + (−4.58 − 6.55i)8-s + (13.2 − 2.72i)10-s + (4.13 − 7.16i)11-s + (7.66 − 4.42i)13-s + (5.38 + 6.05i)14-s + (4.57 + 15.3i)16-s + 28.6·17-s − 7.93·19-s + (−26.8 − 3.14i)20-s + (−12.3 + 11.0i)22-s + (25.3 − 14.6i)23-s + ⋯
L(s)  = 1  + (−0.949 − 0.314i)2-s + (0.801 + 0.597i)4-s + (−1.16 + 0.675i)5-s + (−0.501 − 0.289i)7-s + (−0.572 − 0.819i)8-s + (1.32 − 0.272i)10-s + (0.376 − 0.651i)11-s + (0.589 − 0.340i)13-s + (0.384 + 0.432i)14-s + (0.285 + 0.958i)16-s + 1.68·17-s − 0.417·19-s + (−1.34 − 0.157i)20-s + (−0.562 + 0.500i)22-s + (1.10 − 0.635i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.587 + 0.809i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.587 + 0.809i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.654757 - 0.333787i\)
\(L(\frac12)\) \(\approx\) \(0.654757 - 0.333787i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.89 + 0.629i)T \)
3 \( 1 \)
good5 \( 1 + (5.84 - 3.37i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (3.50 + 2.02i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-4.13 + 7.16i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.66 + 4.42i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 28.6T + 289T^{2} \)
19 \( 1 + 7.93T + 361T^{2} \)
23 \( 1 + (-25.3 + 14.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (15.7 + 9.08i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-40.8 + 23.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 13.3iT - 1.36e3T^{2} \)
41 \( 1 + (31.0 + 53.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-26.5 + 46.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-12.8 - 7.43i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 100. iT - 2.80e3T^{2} \)
59 \( 1 + (-11.4 - 19.9i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (51.3 + 29.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-10.3 - 17.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 54.7iT - 5.04e3T^{2} \)
73 \( 1 + 27.3T + 5.32e3T^{2} \)
79 \( 1 + (-62.9 - 36.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (4.34 - 7.53i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 28.2T + 7.92e3T^{2} \)
97 \( 1 + (0.200 - 0.348i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.74299869832734982830215294312, −10.86468433543463474284224704168, −10.19284507213943746263357183472, −8.915904328351308825028372004308, −7.967933778564543068154280194877, −7.17453726623012549050265823787, −6.09097922490986905245137620046, −3.81623089483409879945778819743, −3.04487185527646729622787773212, −0.68967144607779312716485425353, 1.17133042050787793592835165376, 3.34278501399268079000358852210, 4.93096186567404604738163797851, 6.32517536853320609645802428595, 7.43833954106956277222640748311, 8.272153864636004397610260412932, 9.174787779296375570900971873784, 10.05459023648176166706301893486, 11.33612328732875177236584891960, 12.01315071076787249700117884130

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