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Properties

Label 2-6e3-72.5-c2-0-13
Degree $2$
Conductor $216$
Sign $-0.997 + 0.0739i$
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  + (−1.50 − 1.31i)2-s + (0.519 + 3.96i)4-s + (−3.47 + 6.02i)5-s + (2.29 + 3.97i)7-s + (4.45 − 6.64i)8-s + (13.1 − 4.46i)10-s + (−7.77 − 13.4i)11-s + (−19.3 − 11.1i)13-s + (1.79 − 9.00i)14-s + (−15.4 + 4.12i)16-s − 9.17i·17-s − 4.53i·19-s + (−25.6 − 10.6i)20-s + (−6.07 + 30.4i)22-s + (2.69 + 1.55i)23-s + ⋯
L(s)  = 1  + (−0.751 − 0.659i)2-s + (0.129 + 0.991i)4-s + (−0.695 + 1.20i)5-s + (0.327 + 0.567i)7-s + (0.556 − 0.830i)8-s + (1.31 − 0.446i)10-s + (−0.706 − 1.22i)11-s + (−1.49 − 0.860i)13-s + (0.128 − 0.643i)14-s + (−0.966 + 0.257i)16-s − 0.539i·17-s − 0.238i·19-s + (−1.28 − 0.532i)20-s + (−0.276 + 1.38i)22-s + (0.117 + 0.0676i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0739i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.997 + 0.0739i$
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: \((2,\ 216,\ (\ :1),\ -0.997 + 0.0739i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00300613 - 0.0811766i\)
\(L(\frac12)\) \(\approx\) \(0.00300613 - 0.0811766i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.50 + 1.31i)T \)
3 \( 1 \)
good5 \( 1 + (3.47 - 6.02i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-2.29 - 3.97i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (7.77 + 13.4i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (19.3 + 11.1i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 9.17iT - 289T^{2} \)
19 \( 1 + 4.53iT - 361T^{2} \)
23 \( 1 + (-2.69 - 1.55i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-6.42 - 11.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (2.17 - 3.77i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 23.0iT - 1.36e3T^{2} \)
41 \( 1 + (60.6 + 35.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (51.7 - 29.8i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (32.1 - 18.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 24.3T + 2.80e3T^{2} \)
59 \( 1 + (20.7 - 35.9i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-34.6 + 19.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-24.3 - 14.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 59.8iT - 5.04e3T^{2} \)
73 \( 1 + 53.8T + 5.32e3T^{2} \)
79 \( 1 + (0.557 + 0.965i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-39.8 - 69.0i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 10.6iT - 7.92e3T^{2} \)
97 \( 1 + (72.6 + 125. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.43886654408467103263971766659, −10.71857726085660067498865178768, −9.898646607110278230559817017576, −8.564665828869008972691020241240, −7.75199948508189793371990995896, −6.89365766741819481132605112822, −5.17754453387076279746687114303, −3.31585214107209022120878476817, −2.59536772111803435236211839416, −0.05504025784280158865895910020, 1.77245403397385880801639960228, 4.53409069818284032165470086066, 5.01261984162951409057481682380, 6.82067577627752564158457516979, 7.69702270339289185270301197308, 8.407108991245920780344442322071, 9.592522378970894765196416817429, 10.25898296118762424770209493331, 11.64387150394242384829197157290, 12.42292110086977281086940837890

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