↑

Properties

Label 2-3e6-243.103-c1-0-18
Degree 22
Conductor 729729
Sign −0.430+0.902i-0.430 + 0.902i
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  + (−0.450 − 0.125i)2-s + (−1.52 − 0.920i)4-s + (1.87 − 0.0727i)5-s + (−1.75 − 0.275i)7-s + (1.21 + 1.28i)8-s + (−0.852 − 0.202i)10-s + (−0.122 − 0.894i)11-s + (1.98 − 2.89i)13-s + (0.757 + 0.344i)14-s + (1.27 + 2.42i)16-s + (−1.35 + 0.678i)17-s + (−0.248 − 4.26i)19-s + (−2.92 − 1.61i)20-s + (−0.0570 + 0.417i)22-s + (−0.323 + 0.836i)23-s + ⋯
L(s)  = 1  + (−0.318 − 0.0885i)2-s + (−0.762 − 0.460i)4-s + (0.837 − 0.0325i)5-s + (−0.665 − 0.104i)7-s + (0.428 + 0.454i)8-s + (−0.269 − 0.0638i)10-s + (−0.0368 − 0.269i)11-s + (0.551 − 0.803i)13-s + (0.202 + 0.0920i)14-s + (0.319 + 0.605i)16-s + (−0.327 + 0.164i)17-s + (−0.0570 − 0.979i)19-s + (−0.654 − 0.360i)20-s + (−0.0121 + 0.0891i)22-s + (−0.0673 + 0.174i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((−0.430+0.902i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((−0.430+0.902i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: −0.430+0.902i-0.430 + 0.902i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(685,⋅)\chi_{729} (685, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), −0.430+0.902i)(2,\ 729,\ (\ :1/2),\ -0.430 + 0.902i)

Particular Values

L(1)L(1) ≈\approx 0.457816−0.725657i0.457816 - 0.725657i
L(12)L(\frac12) ≈\approx 0.457816−0.725657i0.457816 - 0.725657i
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+(0.450+0.125i)T+(1.71+1.03i)T2 1 + (0.450 + 0.125i)T + (1.71 + 1.03i)T^{2}
5 1+(−1.87+0.0727i)T+(4.98−0.387i)T2 1 + (-1.87 + 0.0727i)T + (4.98 - 0.387i)T^{2}
7 1+(1.75+0.275i)T+(6.66+2.13i)T2 1 + (1.75 + 0.275i)T + (6.66 + 2.13i)T^{2}
11 1+(0.122+0.894i)T+(−10.5+2.94i)T2 1 + (0.122 + 0.894i)T + (-10.5 + 2.94i)T^{2}
13 1+(−1.98+2.89i)T+(−4.68−12.1i)T2 1 + (-1.98 + 2.89i)T + (-4.68 - 12.1i)T^{2}
17 1+(1.35−0.678i)T+(10.1−13.6i)T2 1 + (1.35 - 0.678i)T + (10.1 - 13.6i)T^{2}
19 1+(0.248+4.26i)T+(−18.8+2.20i)T2 1 + (0.248 + 4.26i)T + (-18.8 + 2.20i)T^{2}
23 1+(0.323−0.836i)T+(−17.0−15.4i)T2 1 + (0.323 - 0.836i)T + (-17.0 - 15.4i)T^{2}
29 1+(5.35+3.82i)T+(9.38+27.4i)T2 1 + (5.35 + 3.82i)T + (9.38 + 27.4i)T^{2}
31 1+(−6.83+7.83i)T+(−4.19−30.7i)T2 1 + (-6.83 + 7.83i)T + (-4.19 - 30.7i)T^{2}
37 1+(4.40+5.92i)T+(−10.6+35.4i)T2 1 + (4.40 + 5.92i)T + (-10.6 + 35.4i)T^{2}
41 1+(1.34−5.21i)T+(−35.8−19.8i)T2 1 + (1.34 - 5.21i)T + (-35.8 - 19.8i)T^{2}
43 1+(−2.48−2.25i)T+(4.16+42.7i)T2 1 + (-2.48 - 2.25i)T + (4.16 + 42.7i)T^{2}
47 1+(5.78+6.62i)T+(−6.36+46.5i)T2 1 + (5.78 + 6.62i)T + (-6.36 + 46.5i)T^{2}
53 1+(0.445−2.52i)T+(−49.8−18.1i)T2 1 + (0.445 - 2.52i)T + (-49.8 - 18.1i)T^{2}
59 1+(−7.15+2.92i)T+(42.1−41.3i)T2 1 + (-7.15 + 2.92i)T + (42.1 - 41.3i)T^{2}
61 1+(4.04−2.44i)T+(28.4−53.9i)T2 1 + (4.04 - 2.44i)T + (28.4 - 53.9i)T^{2}
67 1+(−4.78+3.41i)T+(21.6−63.3i)T2 1 + (-4.78 + 3.41i)T + (21.6 - 63.3i)T^{2}
71 1+(4.36+14.5i)T+(−59.3+39.0i)T2 1 + (4.36 + 14.5i)T + (-59.3 + 39.0i)T^{2}
73 1+(4.63−1.09i)T+(65.2−32.7i)T2 1 + (4.63 - 1.09i)T + (65.2 - 32.7i)T^{2}
79 1+(−5.00+4.90i)T+(1.53−78.9i)T2 1 + (-5.00 + 4.90i)T + (1.53 - 78.9i)T^{2}
83 1+(1.78+6.91i)T+(−72.6+40.0i)T2 1 + (1.78 + 6.91i)T + (-72.6 + 40.0i)T^{2}
89 1+(1.01−3.39i)T+(−74.3−48.9i)T2 1 + (1.01 - 3.39i)T + (-74.3 - 48.9i)T^{2}
97 1+(4.63+0.179i)T+(96.7+7.51i)T2 1 + (4.63 + 0.179i)T + (96.7 + 7.51i)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

−9.921706562628135931617694322962, −9.444696411047764457895258948354, −8.609821165136678093699215287945, −7.69612108127159547572872552957, −6.29875880663135150424207117240, −5.78195507095716785284494746646, −4.72598917495373134208309255061, −3.51992837045701036476445584282, −2.08691606947212474387568509216, −0.51013883769395071071854490527, 1.62390830457342815034613998165, 3.17673084279204489929598248573, 4.20151715298541248980610021248, 5.30739414734265753875624108047, 6.36007949133018508736444189744, 7.13688619283513833930512235396, 8.355030345952157493722511716281, 8.958274681327156904561569303220, 9.817567431390145135755999568286, 10.23712833583706387094930274684

Graph of the ZZ-function along the critical line