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Properties

Label 2-3e6-81.67-c1-0-10
Degree 22
Conductor 729729
Sign 0.328−0.944i0.328 - 0.944i
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  + (−1.29 + 0.848i)2-s + (0.152 − 0.354i)4-s + (0.349 − 0.370i)5-s + (−3.96 − 0.463i)7-s + (−0.432 − 2.45i)8-s + (−0.136 + 0.774i)10-s + (1.09 − 3.66i)11-s + (−0.181 + 3.10i)13-s + (5.50 − 2.76i)14-s + (3.17 + 3.36i)16-s + (−1.41 + 0.513i)17-s + (6.30 + 2.29i)19-s + (−0.0778 − 0.180i)20-s + (1.69 + 5.66i)22-s + (1.17 − 0.137i)23-s + ⋯
L(s)  = 1  + (−0.912 + 0.600i)2-s + (0.0764 − 0.177i)4-s + (0.156 − 0.165i)5-s + (−1.49 − 0.175i)7-s + (−0.153 − 0.868i)8-s + (−0.0431 + 0.244i)10-s + (0.331 − 1.10i)11-s + (−0.0502 + 0.862i)13-s + (1.47 − 0.738i)14-s + (0.793 + 0.840i)16-s + (−0.342 + 0.124i)17-s + (1.44 + 0.526i)19-s + (−0.0173 − 0.0403i)20-s + (0.361 + 1.20i)22-s + (0.245 − 0.0287i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.328−0.944i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.328−0.944i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.328−0.944i0.328 - 0.944i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(28,⋅)\chi_{729} (28, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.328−0.944i)(2,\ 729,\ (\ :1/2),\ 0.328 - 0.944i)

Particular Values

L(1)L(1) ≈\approx 0.577240+0.410600i0.577240 + 0.410600i
L(12)L(\frac12) ≈\approx 0.577240+0.410600i0.577240 + 0.410600i
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+(1.29−0.848i)T+(0.792−1.83i)T2 1 + (1.29 - 0.848i)T + (0.792 - 1.83i)T^{2}
5 1+(−0.349+0.370i)T+(−0.290−4.99i)T2 1 + (-0.349 + 0.370i)T + (-0.290 - 4.99i)T^{2}
7 1+(3.96+0.463i)T+(6.81+1.61i)T2 1 + (3.96 + 0.463i)T + (6.81 + 1.61i)T^{2}
11 1+(−1.09+3.66i)T+(−9.19−6.04i)T2 1 + (-1.09 + 3.66i)T + (-9.19 - 6.04i)T^{2}
13 1+(0.181−3.10i)T+(−12.9−1.50i)T2 1 + (0.181 - 3.10i)T + (-12.9 - 1.50i)T^{2}
17 1+(1.41−0.513i)T+(13.0−10.9i)T2 1 + (1.41 - 0.513i)T + (13.0 - 10.9i)T^{2}
19 1+(−6.30−2.29i)T+(14.5+12.2i)T2 1 + (-6.30 - 2.29i)T + (14.5 + 12.2i)T^{2}
23 1+(−1.17+0.137i)T+(22.3−5.30i)T2 1 + (-1.17 + 0.137i)T + (22.3 - 5.30i)T^{2}
29 1+(−6.17−3.10i)T+(17.3+23.2i)T2 1 + (-6.17 - 3.10i)T + (17.3 + 23.2i)T^{2}
31 1+(0.0276+0.0371i)T+(−8.89+29.6i)T2 1 + (0.0276 + 0.0371i)T + (-8.89 + 29.6i)T^{2}
37 1+(2.33+1.96i)T+(6.42+36.4i)T2 1 + (2.33 + 1.96i)T + (6.42 + 36.4i)T^{2}
41 1+(−4.96−3.26i)T+(16.2+37.6i)T2 1 + (-4.96 - 3.26i)T + (16.2 + 37.6i)T^{2}
43 1+(−0.231+0.0549i)T+(38.4−19.2i)T2 1 + (-0.231 + 0.0549i)T + (38.4 - 19.2i)T^{2}
47 1+(2.82−3.80i)T+(−13.4−45.0i)T2 1 + (2.82 - 3.80i)T + (-13.4 - 45.0i)T^{2}
53 1+(−6.81+11.7i)T+(−26.5−45.8i)T2 1 + (-6.81 + 11.7i)T + (-26.5 - 45.8i)T^{2}
59 1+(−0.400−1.33i)T+(−49.2+32.4i)T2 1 + (-0.400 - 1.33i)T + (-49.2 + 32.4i)T^{2}
61 1+(0.124+0.288i)T+(−41.8+44.3i)T2 1 + (0.124 + 0.288i)T + (-41.8 + 44.3i)T^{2}
67 1+(−4.90+2.46i)T+(40.0−53.7i)T2 1 + (-4.90 + 2.46i)T + (40.0 - 53.7i)T^{2}
71 1+(2.03−11.5i)T+(−66.7−24.2i)T2 1 + (2.03 - 11.5i)T + (-66.7 - 24.2i)T^{2}
73 1+(−2.70−15.3i)T+(−68.5+24.9i)T2 1 + (-2.70 - 15.3i)T + (-68.5 + 24.9i)T^{2}
79 1+(−11.6+7.69i)T+(31.2−72.5i)T2 1 + (-11.6 + 7.69i)T + (31.2 - 72.5i)T^{2}
83 1+(0.587−0.386i)T+(32.8−76.2i)T2 1 + (0.587 - 0.386i)T + (32.8 - 76.2i)T^{2}
89 1+(−2.00−11.3i)T+(−83.6+30.4i)T2 1 + (-2.00 - 11.3i)T + (-83.6 + 30.4i)T^{2}
97 1+(−8.92−9.46i)T+(−5.64+96.8i)T2 1 + (-8.92 - 9.46i)T + (-5.64 + 96.8i)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

−10.14878939121849073566447789385, −9.457562219372637542271092173363, −8.991226617557949414717542001583, −8.077433547483631648061023479813, −6.96247975849617207035647453236, −6.53766883216132873629842728288, −5.51442876694796083393115674679, −3.86445366444088374385612445330, −3.12117737889392918575579307217, −0.957978362628039778226041428453, 0.68034864862434686182500903644, 2.37436433902276399903524737471, 3.20284421553495849932047672267, 4.76566705437931413533895344189, 5.87839612036789466636208568986, 6.81365913345522778426581969265, 7.74592950064278293478759835626, 8.906394101880943073354264445331, 9.545160820612522117273584857005, 10.06986251273376368386458153063

Graph of the ZZ-function along the critical line