↑

Properties

Label 2-3e6-243.103-c1-0-19
Degree 22
Conductor 729729
Sign 0.422+0.906i0.422 + 0.906i
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  + (1.11 + 0.309i)2-s + (−0.574 − 0.346i)4-s + (−2.97 + 0.115i)5-s + (4.35 + 0.681i)7-s + (−2.11 − 2.23i)8-s + (−3.33 − 0.790i)10-s + (−0.228 − 1.67i)11-s + (0.555 − 0.809i)13-s + (4.62 + 2.10i)14-s + (−1.02 − 1.95i)16-s + (0.733 − 0.368i)17-s + (−0.468 − 8.04i)19-s + (1.74 + 0.963i)20-s + (0.263 − 1.92i)22-s + (2.88 − 7.48i)23-s + ⋯
L(s)  = 1  + (0.785 + 0.218i)2-s + (−0.287 − 0.173i)4-s + (−1.32 + 0.0515i)5-s + (1.64 + 0.257i)7-s + (−0.747 − 0.791i)8-s + (−1.05 − 0.249i)10-s + (−0.0688 − 0.504i)11-s + (0.153 − 0.224i)13-s + (1.23 + 0.562i)14-s + (−0.257 − 0.488i)16-s + (0.177 − 0.0893i)17-s + (−0.107 − 1.84i)19-s + (0.390 + 0.215i)20-s + (0.0561 − 0.410i)22-s + (0.602 − 1.56i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.422+0.906i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.422+0.906i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.422+0.906i0.422 + 0.906i
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), 0.422+0.906i)(2,\ 729,\ (\ :1/2),\ 0.422 + 0.906i)

Particular Values

L(1)L(1) ≈\approx 1.35049−0.860450i1.35049 - 0.860450i
L(12)L(\frac12) ≈\approx 1.35049−0.860450i1.35049 - 0.860450i
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+(−1.11−0.309i)T+(1.71+1.03i)T2 1 + (-1.11 - 0.309i)T + (1.71 + 1.03i)T^{2}
5 1+(2.97−0.115i)T+(4.98−0.387i)T2 1 + (2.97 - 0.115i)T + (4.98 - 0.387i)T^{2}
7 1+(−4.35−0.681i)T+(6.66+2.13i)T2 1 + (-4.35 - 0.681i)T + (6.66 + 2.13i)T^{2}
11 1+(0.228+1.67i)T+(−10.5+2.94i)T2 1 + (0.228 + 1.67i)T + (-10.5 + 2.94i)T^{2}
13 1+(−0.555+0.809i)T+(−4.68−12.1i)T2 1 + (-0.555 + 0.809i)T + (-4.68 - 12.1i)T^{2}
17 1+(−0.733+0.368i)T+(10.1−13.6i)T2 1 + (-0.733 + 0.368i)T + (10.1 - 13.6i)T^{2}
19 1+(0.468+8.04i)T+(−18.8+2.20i)T2 1 + (0.468 + 8.04i)T + (-18.8 + 2.20i)T^{2}
23 1+(−2.88+7.48i)T+(−17.0−15.4i)T2 1 + (-2.88 + 7.48i)T + (-17.0 - 15.4i)T^{2}
29 1+(−2.38−1.70i)T+(9.38+27.4i)T2 1 + (-2.38 - 1.70i)T + (9.38 + 27.4i)T^{2}
31 1+(1.81−2.08i)T+(−4.19−30.7i)T2 1 + (1.81 - 2.08i)T + (-4.19 - 30.7i)T^{2}
37 1+(1.78+2.39i)T+(−10.6+35.4i)T2 1 + (1.78 + 2.39i)T + (-10.6 + 35.4i)T^{2}
41 1+(−2.19+8.50i)T+(−35.8−19.8i)T2 1 + (-2.19 + 8.50i)T + (-35.8 - 19.8i)T^{2}
43 1+(1.13+1.02i)T+(4.16+42.7i)T2 1 + (1.13 + 1.02i)T + (4.16 + 42.7i)T^{2}
47 1+(−1.86−2.13i)T+(−6.36+46.5i)T2 1 + (-1.86 - 2.13i)T + (-6.36 + 46.5i)T^{2}
53 1+(1.22−6.95i)T+(−49.8−18.1i)T2 1 + (1.22 - 6.95i)T + (-49.8 - 18.1i)T^{2}
59 1+(−2.23+0.913i)T+(42.1−41.3i)T2 1 + (-2.23 + 0.913i)T + (42.1 - 41.3i)T^{2}
61 1+(1.90−1.14i)T+(28.4−53.9i)T2 1 + (1.90 - 1.14i)T + (28.4 - 53.9i)T^{2}
67 1+(7.25−5.18i)T+(21.6−63.3i)T2 1 + (7.25 - 5.18i)T + (21.6 - 63.3i)T^{2}
71 1+(2.04+6.83i)T+(−59.3+39.0i)T2 1 + (2.04 + 6.83i)T + (-59.3 + 39.0i)T^{2}
73 1+(−14.6+3.47i)T+(65.2−32.7i)T2 1 + (-14.6 + 3.47i)T + (65.2 - 32.7i)T^{2}
79 1+(10.8−10.6i)T+(1.53−78.9i)T2 1 + (10.8 - 10.6i)T + (1.53 - 78.9i)T^{2}
83 1+(1.54+6.00i)T+(−72.6+40.0i)T2 1 + (1.54 + 6.00i)T + (-72.6 + 40.0i)T^{2}
89 1+(5.19−17.3i)T+(−74.3−48.9i)T2 1 + (5.19 - 17.3i)T + (-74.3 - 48.9i)T^{2}
97 1+(−13.4−0.522i)T+(96.7+7.51i)T2 1 + (-13.4 - 0.522i)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

−10.68953112616592275826788713469, −8.959127549274635581023788409763, −8.590602296841836995237789012840, −7.60448696983116287694939132371, −6.72268813347388002859410220487, −5.38931810581654658522080635992, −4.74057305784342526530250512611, −4.05655378208675481332578367593, −2.79801724103107405701565754992, −0.70551847543815980067025588361, 1.64855448895931878419748270619, 3.39258958823643269161813833602, 4.16032889863781852291340132422, 4.80429602769727810877586250548, 5.73953633878273622231144072573, 7.36554932786645450144687304933, 8.012970875632591890884548603101, 8.448592204933596627779829310906, 9.740754130797531053356741684235, 10.95446102836477320280404918705

Graph of the ZZ-function along the critical line