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Properties

Label 2-3e6-9.7-c1-0-20
Degree 22
Conductor 729729
Sign 0.5+0.866i0.5 + 0.866i
Analytic cond. 5.821095.82109
Root an. cond. 2.412692.41269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.207 − 0.359i)2-s + (0.913 + 1.58i)4-s + (−1.10 − 1.91i)5-s + (0.659 − 1.14i)7-s + 1.59·8-s − 0.920·10-s + (2.60 − 4.51i)11-s + (0.00902 + 0.0156i)13-s + (−0.274 − 0.474i)14-s + (−1.49 + 2.59i)16-s − 3.13·17-s + 0.417·19-s + (2.02 − 3.50i)20-s + (−1.08 − 1.87i)22-s + (0.517 + 0.895i)23-s + ⋯
L(s)  = 1  + (0.146 − 0.254i)2-s + (0.456 + 0.791i)4-s + (−0.495 − 0.857i)5-s + (0.249 − 0.431i)7-s + 0.562·8-s − 0.291·10-s + (0.786 − 1.36i)11-s + (0.00250 + 0.00433i)13-s + (−0.0732 − 0.126i)14-s + (−0.374 + 0.648i)16-s − 0.759·17-s + 0.0957·19-s + (0.452 − 0.783i)20-s + (−0.230 − 0.400i)22-s + (0.107 + 0.186i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.5+0.866i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.5+0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.5+0.866i0.5 + 0.866i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(487,⋅)\chi_{729} (487, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.5+0.866i)(2,\ 729,\ (\ :1/2),\ 0.5 + 0.866i)

Particular Values

L(1)L(1) ≈\approx 1.51943−0.877246i1.51943 - 0.877246i
L(12)L(\frac12) ≈\approx 1.51943−0.877246i1.51943 - 0.877246i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
good2 1+(−0.207+0.359i)T+(−1−1.73i)T2 1 + (-0.207 + 0.359i)T + (-1 - 1.73i)T^{2}
5 1+(1.10+1.91i)T+(−2.5+4.33i)T2 1 + (1.10 + 1.91i)T + (-2.5 + 4.33i)T^{2}
7 1+(−0.659+1.14i)T+(−3.5−6.06i)T2 1 + (-0.659 + 1.14i)T + (-3.5 - 6.06i)T^{2}
11 1+(−2.60+4.51i)T+(−5.5−9.52i)T2 1 + (-2.60 + 4.51i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.00902−0.0156i)T+(−6.5+11.2i)T2 1 + (-0.00902 - 0.0156i)T + (-6.5 + 11.2i)T^{2}
17 1+3.13T+17T2 1 + 3.13T + 17T^{2}
19 1−0.417T+19T2 1 - 0.417T + 19T^{2}
23 1+(−0.517−0.895i)T+(−11.5+19.9i)T2 1 + (-0.517 - 0.895i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.90+6.76i)T+(−14.5−25.1i)T2 1 + (-3.90 + 6.76i)T + (-14.5 - 25.1i)T^{2}
31 1+(1.86+3.22i)T+(−15.5+26.8i)T2 1 + (1.86 + 3.22i)T + (-15.5 + 26.8i)T^{2}
37 1−4.42T+37T2 1 - 4.42T + 37T^{2}
41 1+(−1.83−3.18i)T+(−20.5+35.5i)T2 1 + (-1.83 - 3.18i)T + (-20.5 + 35.5i)T^{2}
43 1+(−4.15+7.19i)T+(−21.5−37.2i)T2 1 + (-4.15 + 7.19i)T + (-21.5 - 37.2i)T^{2}
47 1+(−3.54+6.14i)T+(−23.5−40.7i)T2 1 + (-3.54 + 6.14i)T + (-23.5 - 40.7i)T^{2}
53 1+1.30T+53T2 1 + 1.30T + 53T^{2}
59 1+(−1.85−3.20i)T+(−29.5+51.0i)T2 1 + (-1.85 - 3.20i)T + (-29.5 + 51.0i)T^{2}
61 1+(3.45−5.98i)T+(−30.5−52.8i)T2 1 + (3.45 - 5.98i)T + (-30.5 - 52.8i)T^{2}
67 1+(−5.51−9.54i)T+(−33.5+58.0i)T2 1 + (-5.51 - 9.54i)T + (-33.5 + 58.0i)T^{2}
71 1+6.08T+71T2 1 + 6.08T + 71T^{2}
73 1+0.546T+73T2 1 + 0.546T + 73T^{2}
79 1+(0.244−0.423i)T+(−39.5−68.4i)T2 1 + (0.244 - 0.423i)T + (-39.5 - 68.4i)T^{2}
83 1+(2.30−3.99i)T+(−41.5−71.8i)T2 1 + (2.30 - 3.99i)T + (-41.5 - 71.8i)T^{2}
89 1+3.37T+89T2 1 + 3.37T + 89T^{2}
97 1+(4.97−8.60i)T+(−48.5−84.0i)T2 1 + (4.97 - 8.60i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.50405198072719088568369464353, −9.148393243024875325243673631237, −8.474588929997743560379054485414, −7.81913107369866811151029452869, −6.82472078853620657370317940369, −5.80440640999260606844842058324, −4.35500284011603084061215121581, −3.90618598406839677672918116741, −2.58996885623054026515537616075, −0.942077806082286805687160254170, 1.62941111586927446646497615605, 2.79064842152111645317478439527, 4.26107663600288145524046492480, 5.12027780206802657918276976746, 6.35327455548472327451285745488, 6.92828468549285061431627344546, 7.61508943466669773993929393038, 8.945896919982508547534742006332, 9.728884522265365264897247460505, 10.70489032839915050723185866519

Graph of the ZZ-function along the critical line