Properties

Label 2-6e3-216.101-c2-0-28
Degree $2$
Conductor $216$
Sign $-0.760 - 0.649i$
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  + (1.46 + 1.36i)2-s + (1.21 + 2.74i)3-s + (0.299 + 3.98i)4-s + (0.719 + 4.07i)5-s + (−1.95 + 5.67i)6-s + (0.106 + 0.0891i)7-s + (−4.98 + 6.25i)8-s + (−6.07 + 6.64i)9-s + (−4.49 + 6.96i)10-s + (3.25 − 18.4i)11-s + (−10.5 + 5.64i)12-s + (5.32 − 14.6i)13-s + (0.0344 + 0.275i)14-s + (−10.3 + 6.91i)15-s + (−15.8 + 2.38i)16-s + (−1.59 + 0.922i)17-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)2-s + (0.403 + 0.915i)3-s + (0.0748 + 0.997i)4-s + (0.143 + 0.815i)5-s + (−0.326 + 0.945i)6-s + (0.0151 + 0.0127i)7-s + (−0.623 + 0.781i)8-s + (−0.674 + 0.738i)9-s + (−0.449 + 0.696i)10-s + (0.296 − 1.68i)11-s + (−0.882 + 0.470i)12-s + (0.409 − 1.12i)13-s + (0.00246 + 0.0196i)14-s + (−0.688 + 0.460i)15-s + (−0.988 + 0.149i)16-s + (−0.0939 + 0.0542i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.760 - 0.649i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.760 - 0.649i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.878331 + 2.38001i\)
\(L(\frac12)\) \(\approx\) \(0.878331 + 2.38001i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.46 - 1.36i)T \)
3 \( 1 + (-1.21 - 2.74i)T \)
good5 \( 1 + (-0.719 - 4.07i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (-0.106 - 0.0891i)T + (8.50 + 48.2i)T^{2} \)
11 \( 1 + (-3.25 + 18.4i)T + (-113. - 41.3i)T^{2} \)
13 \( 1 + (-5.32 + 14.6i)T + (-129. - 108. i)T^{2} \)
17 \( 1 + (1.59 - 0.922i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-25.5 - 14.7i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-12.8 - 15.3i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (14.9 - 5.43i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (44.5 - 37.4i)T + (166. - 946. i)T^{2} \)
37 \( 1 + (-15.6 + 9.02i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-9.34 + 25.6i)T + (-1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-70.2 - 12.3i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (31.7 - 37.8i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 + 52.5T + 2.80e3T^{2} \)
59 \( 1 + (-4.51 - 25.6i)T + (-3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-34.3 + 40.8i)T + (-646. - 3.66e3i)T^{2} \)
67 \( 1 + (-27.2 + 74.8i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (6.09 - 3.51i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (5.25 - 9.10i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-68.1 + 24.7i)T + (4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (-24.4 + 8.90i)T + (5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (7.40 + 4.27i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-13.1 + 74.5i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−12.72211635759963003452387086933, −11.21575275751127782838188759740, −10.83884961847322160876863472329, −9.367071922832813607500558917963, −8.388781674768953694160972179426, −7.44361294283399161923642067922, −5.96892173037099720600154013430, −5.31555346417275987500504534607, −3.51205210413783938536082482814, −3.15276704251990918741018972964, 1.22987083239894465127637820592, 2.36514306973337367808988327294, 4.07775047062945409420917974103, 5.17419750252445478986998578941, 6.55873693509433349857103936704, 7.45332519914347284410072118444, 9.229511606987401185159528715692, 9.403315384932278459699738390207, 11.18775996999034195369291799534, 11.94751239480183838964208699866

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