↑

Properties

Label 2-720-45.34-c1-0-21
Degree 22
Conductor 720720
Sign 0.999+0.00269i0.999 + 0.00269i
Analytic cond. 5.749225.74922
Root an. cond. 2.397752.39775
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  + (−0.690 + 1.58i)3-s + (1.99 − 1.00i)5-s + (1.11 + 0.644i)7-s + (−2.04 − 2.19i)9-s + (2.54 − 4.41i)11-s + (3.09 − 1.78i)13-s + (0.225 + 3.86i)15-s − 0.895i·17-s − 5.34·19-s + (−1.79 + 1.32i)21-s + (4.38 − 2.53i)23-s + (2.96 − 4.02i)25-s + (4.89 − 1.73i)27-s + (−1.5 + 2.59i)29-s + (−3.29 − 5.71i)31-s + ⋯
L(s)  = 1  + (−0.398 + 0.917i)3-s + (0.892 − 0.451i)5-s + (0.421 + 0.243i)7-s + (−0.682 − 0.731i)9-s + (0.767 − 1.32i)11-s + (0.857 − 0.495i)13-s + (0.0582 + 0.998i)15-s − 0.217i·17-s − 1.22·19-s + (−0.391 + 0.289i)21-s + (0.914 − 0.528i)23-s + (0.592 − 0.805i)25-s + (0.942 − 0.334i)27-s + (−0.278 + 0.482i)29-s + (−0.592 − 1.02i)31-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.999+0.00269i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00269i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1/2) L(s)=((0.999+0.00269i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00269i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.999+0.00269i0.999 + 0.00269i
Analytic conductor: 5.749225.74922
Root analytic conductor: 2.397752.39775
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ720(529,⋅)\chi_{720} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1/2), 0.999+0.00269i)(2,\ 720,\ (\ :1/2),\ 0.999 + 0.00269i)

Particular Values

L(1)L(1) ≈\approx 1.66214−0.00223641i1.66214 - 0.00223641i
L(12)L(\frac12) ≈\approx 1.66214−0.00223641i1.66214 - 0.00223641i
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)
bad2 1 1
3 1+(0.690−1.58i)T 1 + (0.690 - 1.58i)T
5 1+(−1.99+1.00i)T 1 + (-1.99 + 1.00i)T
good7 1+(−1.11−0.644i)T+(3.5+6.06i)T2 1 + (-1.11 - 0.644i)T + (3.5 + 6.06i)T^{2}
11 1+(−2.54+4.41i)T+(−5.5−9.52i)T2 1 + (-2.54 + 4.41i)T + (-5.5 - 9.52i)T^{2}
13 1+(−3.09+1.78i)T+(6.5−11.2i)T2 1 + (-3.09 + 1.78i)T + (6.5 - 11.2i)T^{2}
17 1+0.895iT−17T2 1 + 0.895iT - 17T^{2}
19 1+5.34T+19T2 1 + 5.34T + 19T^{2}
23 1+(−4.38+2.53i)T+(11.5−19.9i)T2 1 + (-4.38 + 2.53i)T + (11.5 - 19.9i)T^{2}
29 1+(1.5−2.59i)T+(−14.5−25.1i)T2 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2}
31 1+(3.29+5.71i)T+(−15.5+26.8i)T2 1 + (3.29 + 5.71i)T + (-15.5 + 26.8i)T^{2}
37 1−7.24iT−37T2 1 - 7.24iT - 37T^{2}
41 1+(−3.92−6.79i)T+(−20.5+35.5i)T2 1 + (-3.92 - 6.79i)T + (-20.5 + 35.5i)T^{2}
43 1+(−9.46−5.46i)T+(21.5+37.2i)T2 1 + (-9.46 - 5.46i)T + (21.5 + 37.2i)T^{2}
47 1+(2.57+1.48i)T+(23.5+40.7i)T2 1 + (2.57 + 1.48i)T + (23.5 + 40.7i)T^{2}
53 1−4.78iT−53T2 1 - 4.78iT - 53T^{2}
59 1+(−2.87−4.98i)T+(−29.5+51.0i)T2 1 + (-2.87 - 4.98i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.17−3.75i)T+(−30.5−52.8i)T2 1 + (2.17 - 3.75i)T + (-30.5 - 52.8i)T^{2}
67 1+(7.39−4.26i)T+(33.5−58.0i)T2 1 + (7.39 - 4.26i)T + (33.5 - 58.0i)T^{2}
71 1−5.34T+71T2 1 - 5.34T + 71T^{2}
73 1+9.34iT−73T2 1 + 9.34iT - 73T^{2}
79 1+(−0.370+0.642i)T+(−39.5−68.4i)T2 1 + (-0.370 + 0.642i)T + (-39.5 - 68.4i)T^{2}
83 1+(−7.97−4.60i)T+(41.5+71.8i)T2 1 + (-7.97 - 4.60i)T + (41.5 + 71.8i)T^{2}
89 1−9.24T+89T2 1 - 9.24T + 89T^{2}
97 1+(2.99+1.72i)T+(48.5+84.0i)T2 1 + (2.99 + 1.72i)T + (48.5 + 84.0i)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

−10.60288156660340871400052828521, −9.409499135803636294353086448481, −8.882280454155641348565540672193, −8.215554280954697575376804813020, −6.38027138103629406251118231237, −5.97128872463973113769005550388, −5.02789341729405913181015122514, −4.04470122441978125938517679346, −2.84220814642831326813051124062, −1.06003824912197139798975400825, 1.49584832387724757488527027396, 2.22785596311731643663686944464, 3.95253112106624186099734866204, 5.17296551637279303611454337854, 6.17432817085334799244763027898, 6.85362400463265829143770530992, 7.51420063041616895568229391148, 8.778533194788781528604197157783, 9.458689460983131189282003172811, 10.82331855264020598605257325809

Graph of the ZZ-function along the critical line