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Properties

Label 2-6e3-24.5-c2-0-0
Degree $2$
Conductor $216$
Sign $0.182 - 0.983i$
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  + (0.122 − 1.99i)2-s + (−3.97 − 0.487i)4-s − 5.18·5-s + 3.67·7-s + (−1.45 + 7.86i)8-s + (−0.633 + 10.3i)10-s − 12.8·11-s + 8.72i·13-s + (0.448 − 7.33i)14-s + (15.5 + 3.87i)16-s + 18.6i·17-s + 30.7i·19-s + (20.5 + 2.52i)20-s + (−1.57 + 25.7i)22-s − 20.7i·23-s + ⋯
L(s)  = 1  + (0.0610 − 0.998i)2-s + (−0.992 − 0.121i)4-s − 1.03·5-s + 0.524·7-s + (−0.182 + 0.983i)8-s + (−0.0633 + 1.03i)10-s − 1.17·11-s + 0.670i·13-s + (0.0320 − 0.523i)14-s + (0.970 + 0.242i)16-s + 1.09i·17-s + 1.61i·19-s + (1.02 + 0.126i)20-s + (−0.0715 + 1.16i)22-s − 0.903i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.182 - 0.983i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.182 - 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.289421 + 0.240679i\)
\(L(\frac12)\) \(\approx\) \(0.289421 + 0.240679i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.122 + 1.99i)T \)
3 \( 1 \)
good5 \( 1 + 5.18T + 25T^{2} \)
7 \( 1 - 3.67T + 49T^{2} \)
11 \( 1 + 12.8T + 121T^{2} \)
13 \( 1 - 8.72iT - 169T^{2} \)
17 \( 1 - 18.6iT - 289T^{2} \)
19 \( 1 - 30.7iT - 361T^{2} \)
23 \( 1 + 20.7iT - 529T^{2} \)
29 \( 1 + 27.8T + 841T^{2} \)
31 \( 1 + 16.6T + 961T^{2} \)
37 \( 1 + 60.5iT - 1.36e3T^{2} \)
41 \( 1 + 28.2iT - 1.68e3T^{2} \)
43 \( 1 - 51.8iT - 1.84e3T^{2} \)
47 \( 1 - 69.2iT - 2.20e3T^{2} \)
53 \( 1 + 28.5T + 2.80e3T^{2} \)
59 \( 1 + 61.1T + 3.48e3T^{2} \)
61 \( 1 + 105. iT - 3.72e3T^{2} \)
67 \( 1 - 16.9iT - 4.48e3T^{2} \)
71 \( 1 - 26.1iT - 5.04e3T^{2} \)
73 \( 1 - 102.T + 5.32e3T^{2} \)
79 \( 1 + 92.2T + 6.24e3T^{2} \)
83 \( 1 - 36.4T + 6.88e3T^{2} \)
89 \( 1 - 73.5iT - 7.92e3T^{2} \)
97 \( 1 + 186.T + 9.40e3T^{2} \)
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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

−12.40645126542950970015283946798, −11.16386726384114096727954882565, −10.75322432634770190184643658164, −9.539353060695504720908269845193, −8.238827991364914643584641516790, −7.76928789390519004327381863587, −5.80739341270783297924998776833, −4.49349979289060765852229879668, −3.58767127033339052749589373964, −1.91113349922607024981734939889, 0.20289816142045880986014010592, 3.20428697967706399166814048992, 4.68864436716611363848327992760, 5.42944567357982333467759484546, 7.10658507902000102765832494825, 7.70192021844648104720684035015, 8.553955879799998877987581752037, 9.711706737050675891927858156260, 11.00551620866037068127491418587, 11.89207637736100544153750645581

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