↑

Properties

Label 2-3e6-27.16-c1-0-15
Degree 22
Conductor 729729
Sign 0.893−0.448i0.893 - 0.448i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.673 + 0.565i)2-s + (−0.213 − 1.20i)4-s + (3.64 + 1.32i)5-s + (−0.379 + 2.15i)7-s + (1.41 − 2.45i)8-s + (1.70 + 2.95i)10-s + (0.152 − 0.0555i)11-s + (1.84 − 1.55i)13-s + (−1.47 + 1.23i)14-s + (0.0393 − 0.0143i)16-s + (1.5 + 2.59i)17-s + (−1.79 + 3.11i)19-s + (0.826 − 4.68i)20-s + (0.134 + 0.0488i)22-s + (−0.492 − 2.79i)23-s + ⋯
L(s)  = 1  + (0.476 + 0.399i)2-s + (−0.106 − 0.604i)4-s + (1.63 + 0.593i)5-s + (−0.143 + 0.813i)7-s + (0.501 − 0.868i)8-s + (0.539 + 0.934i)10-s + (0.0460 − 0.0167i)11-s + (0.512 − 0.429i)13-s + (−0.393 + 0.330i)14-s + (0.00984 − 0.00358i)16-s + (0.363 + 0.630i)17-s + (−0.412 + 0.714i)19-s + (0.184 − 1.04i)20-s + (0.0286 + 0.0104i)22-s + (−0.102 − 0.582i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.893−0.448i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.893−0.448i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.893−0.448i0.893 - 0.448i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(406,⋅)\chi_{729} (406, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.893−0.448i)(2,\ 729,\ (\ :1/2),\ 0.893 - 0.448i)

Particular Values

L(1)L(1) ≈\approx 2.44891+0.580403i2.44891 + 0.580403i
L(12)L(\frac12) ≈\approx 2.44891+0.580403i2.44891 + 0.580403i
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+(−0.673−0.565i)T+(0.347+1.96i)T2 1 + (-0.673 - 0.565i)T + (0.347 + 1.96i)T^{2}
5 1+(−3.64−1.32i)T+(3.83+3.21i)T2 1 + (-3.64 - 1.32i)T + (3.83 + 3.21i)T^{2}
7 1+(0.379−2.15i)T+(−6.57−2.39i)T2 1 + (0.379 - 2.15i)T + (-6.57 - 2.39i)T^{2}
11 1+(−0.152+0.0555i)T+(8.42−7.07i)T2 1 + (-0.152 + 0.0555i)T + (8.42 - 7.07i)T^{2}
13 1+(−1.84+1.55i)T+(2.25−12.8i)T2 1 + (-1.84 + 1.55i)T + (2.25 - 12.8i)T^{2}
17 1+(−1.5−2.59i)T+(−8.5+14.7i)T2 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.79−3.11i)T+(−9.5−16.4i)T2 1 + (1.79 - 3.11i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.492+2.79i)T+(−21.6+7.86i)T2 1 + (0.492 + 2.79i)T + (-21.6 + 7.86i)T^{2}
29 1+(5.14+4.31i)T+(5.03+28.5i)T2 1 + (5.14 + 4.31i)T + (5.03 + 28.5i)T^{2}
31 1+(0.900+5.10i)T+(−29.1+10.6i)T2 1 + (0.900 + 5.10i)T + (-29.1 + 10.6i)T^{2}
37 1+(−3.31−5.74i)T+(−18.5+32.0i)T2 1 + (-3.31 - 5.74i)T + (-18.5 + 32.0i)T^{2}
41 1+(−4.44+3.72i)T+(7.11−40.3i)T2 1 + (-4.44 + 3.72i)T + (7.11 - 40.3i)T^{2}
43 1+(−5.85+2.12i)T+(32.9−27.6i)T2 1 + (-5.85 + 2.12i)T + (32.9 - 27.6i)T^{2}
47 1+(1.28−7.28i)T+(−44.1−16.0i)T2 1 + (1.28 - 7.28i)T + (-44.1 - 16.0i)T^{2}
53 1+1.40T+53T2 1 + 1.40T + 53T^{2}
59 1+(4.81+1.75i)T+(45.1+37.9i)T2 1 + (4.81 + 1.75i)T + (45.1 + 37.9i)T^{2}
61 1+(0.656−3.72i)T+(−57.3−20.8i)T2 1 + (0.656 - 3.72i)T + (-57.3 - 20.8i)T^{2}
67 1+(4.49−3.76i)T+(11.6−65.9i)T2 1 + (4.49 - 3.76i)T + (11.6 - 65.9i)T^{2}
71 1+(7.65+13.2i)T+(−35.5+61.4i)T2 1 + (7.65 + 13.2i)T + (-35.5 + 61.4i)T^{2}
73 1+(4.34−7.51i)T+(−36.5−63.2i)T2 1 + (4.34 - 7.51i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.971+0.815i)T+(13.7+77.7i)T2 1 + (0.971 + 0.815i)T + (13.7 + 77.7i)T^{2}
83 1+(−6.49−5.44i)T+(14.4+81.7i)T2 1 + (-6.49 - 5.44i)T + (14.4 + 81.7i)T^{2}
89 1+(−3.86+6.68i)T+(−44.5−77.0i)T2 1 + (-3.86 + 6.68i)T + (-44.5 - 77.0i)T^{2}
97 1+(−3.67+1.33i)T+(74.3−62.3i)T2 1 + (-3.67 + 1.33i)T + (74.3 - 62.3i)T^{2}
show more
show less
   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

−10.37149984674044292491271602621, −9.659241432212485288371583358629, −9.008886339504521614017021434275, −7.72300881661240446542896653630, −6.39741585564678329963751217049, −5.95332593013596036257693295062, −5.54030650392550113103358919537, −4.14684791338315242107953857003, −2.65832921717516611532379786473, −1.59003019231981783883332441344, 1.43703667937789102383478970181, 2.60277388037505361012481497282, 3.82905360015540609018045200664, 4.83795533021789518208528383653, 5.65109881310182232145496816693, 6.76616400847717730892397778451, 7.68271369880411180354147403106, 8.968303506005734082332832279343, 9.314167855791265639250028357834, 10.45049552979046521493965176552

Graph of the ZZ-function along the critical line