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Properties

Label 2-3e6-27.22-c1-0-29
Degree 22
Conductor 729729
Sign −0.116+0.993i-0.116 + 0.993i
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  + (1.50 − 1.26i)2-s + (0.326 − 1.85i)4-s + (3.47 − 1.26i)5-s + (−0.407 − 2.31i)7-s + (0.118 + 0.205i)8-s + (3.64 − 6.31i)10-s + (−2.04 − 0.745i)11-s + (−3.61 − 3.03i)13-s + (−3.54 − 2.97i)14-s + (3.97 + 1.44i)16-s + (−1.46 + 2.54i)17-s + (3.11 + 5.39i)19-s + (−1.20 − 6.85i)20-s + (−4.03 + 1.46i)22-s + (−0.0901 + 0.511i)23-s + ⋯
L(s)  = 1  + (1.06 − 0.895i)2-s + (0.163 − 0.925i)4-s + (1.55 − 0.566i)5-s + (−0.154 − 0.873i)7-s + (0.0419 + 0.0727i)8-s + (1.15 − 1.99i)10-s + (−0.617 − 0.224i)11-s + (−1.00 − 0.840i)13-s + (−0.946 − 0.794i)14-s + (0.992 + 0.361i)16-s + (−0.355 + 0.616i)17-s + (0.714 + 1.23i)19-s + (−0.270 − 1.53i)20-s + (−0.859 + 0.312i)22-s + (−0.0187 + 0.106i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((−0.116+0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((−0.116+0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: −0.116+0.993i-0.116 + 0.993i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(325,⋅)\chi_{729} (325, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), −0.116+0.993i)(2,\ 729,\ (\ :1/2),\ -0.116 + 0.993i)

Particular Values

L(1)L(1) ≈\approx 2.12940−2.39279i2.12940 - 2.39279i
L(12)L(\frac12) ≈\approx 2.12940−2.39279i2.12940 - 2.39279i
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+(−1.50+1.26i)T+(0.347−1.96i)T2 1 + (-1.50 + 1.26i)T + (0.347 - 1.96i)T^{2}
5 1+(−3.47+1.26i)T+(3.83−3.21i)T2 1 + (-3.47 + 1.26i)T + (3.83 - 3.21i)T^{2}
7 1+(0.407+2.31i)T+(−6.57+2.39i)T2 1 + (0.407 + 2.31i)T + (-6.57 + 2.39i)T^{2}
11 1+(2.04+0.745i)T+(8.42+7.07i)T2 1 + (2.04 + 0.745i)T + (8.42 + 7.07i)T^{2}
13 1+(3.61+3.03i)T+(2.25+12.8i)T2 1 + (3.61 + 3.03i)T + (2.25 + 12.8i)T^{2}
17 1+(1.46−2.54i)T+(−8.5−14.7i)T2 1 + (1.46 - 2.54i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.11−5.39i)T+(−9.5+16.4i)T2 1 + (-3.11 - 5.39i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.0901−0.511i)T+(−21.6−7.86i)T2 1 + (0.0901 - 0.511i)T + (-21.6 - 7.86i)T^{2}
29 1+(−2.67+2.24i)T+(5.03−28.5i)T2 1 + (-2.67 + 2.24i)T + (5.03 - 28.5i)T^{2}
31 1+(0.747−4.23i)T+(−29.1−10.6i)T2 1 + (0.747 - 4.23i)T + (-29.1 - 10.6i)T^{2}
37 1+(1.20−2.08i)T+(−18.5−32.0i)T2 1 + (1.20 - 2.08i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.91+1.60i)T+(7.11+40.3i)T2 1 + (1.91 + 1.60i)T + (7.11 + 40.3i)T^{2}
43 1+(1+0.363i)T+(32.9+27.6i)T2 1 + (1 + 0.363i)T + (32.9 + 27.6i)T^{2}
47 1+(−0.0412−0.233i)T+(−44.1+16.0i)T2 1 + (-0.0412 - 0.233i)T + (-44.1 + 16.0i)T^{2}
53 1+4.66T+53T2 1 + 4.66T + 53T^{2}
59 1+(12.5−4.55i)T+(45.1−37.9i)T2 1 + (12.5 - 4.55i)T + (45.1 - 37.9i)T^{2}
61 1+(0.638+3.61i)T+(−57.3+20.8i)T2 1 + (0.638 + 3.61i)T + (-57.3 + 20.8i)T^{2}
67 1+(−10.9−9.19i)T+(11.6+65.9i)T2 1 + (-10.9 - 9.19i)T + (11.6 + 65.9i)T^{2}
71 1+(−0.601+1.04i)T+(−35.5−61.4i)T2 1 + (-0.601 + 1.04i)T + (-35.5 - 61.4i)T^{2}
73 1+(−2.34−4.05i)T+(−36.5+63.2i)T2 1 + (-2.34 - 4.05i)T + (-36.5 + 63.2i)T^{2}
79 1+(9.80−8.22i)T+(13.7−77.7i)T2 1 + (9.80 - 8.22i)T + (13.7 - 77.7i)T^{2}
83 1+(−8.65+7.26i)T+(14.4−81.7i)T2 1 + (-8.65 + 7.26i)T + (14.4 - 81.7i)T^{2}
89 1+(−0.349−0.605i)T+(−44.5+77.0i)T2 1 + (-0.349 - 0.605i)T + (-44.5 + 77.0i)T^{2}
97 1+(−6.65−2.42i)T+(74.3+62.3i)T2 1 + (-6.65 - 2.42i)T + (74.3 + 62.3i)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

−10.12739169550036213695942541745, −9.937154224060857342945644024766, −8.515927738857142961638368199229, −7.54565607532432448167081943881, −6.16032367260358192432750980339, −5.39896156623780067430726539298, −4.74751334684969392858139884762, −3.50294710750217477675815095163, −2.45761220700379642066382227535, −1.36235173811013166272104627780, 2.18353243566683180971786423464, 2.98226978638077791889148029052, 4.78715124933758586004548095178, 5.21855512161456302866753910816, 6.17463000750522432559330607215, 6.78180216802383861575655621221, 7.56878110992236554107520164187, 9.165387689340921434782959961874, 9.578696419762578646119724367397, 10.50984938606945278451407932485

Graph of the ZZ-function along the critical line