↑

Properties

Label 2-360-120.59-c1-0-13
Degree 22
Conductor 360360
Sign 0.249−0.968i0.249 - 0.968i
Analytic cond. 2.874612.87461
Root an. cond. 1.695461.69546
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  + (0.927 + 1.06i)2-s + (−0.280 + 1.98i)4-s + (2.18 + 0.468i)5-s + 3.02·7-s + (−2.37 + 1.53i)8-s + (1.52 + 2.76i)10-s − 3.62i·11-s − 1.69·13-s + (2.80 + 3.22i)14-s + (−3.84 − 1.11i)16-s − 6.60·17-s + 5.12·19-s + (−1.54 + 4.19i)20-s + (3.86 − 3.35i)22-s + 6.67i·23-s + ⋯
L(s)  = 1  + (0.655 + 0.755i)2-s + (−0.140 + 0.990i)4-s + (0.977 + 0.209i)5-s + 1.14·7-s + (−0.839 + 0.543i)8-s + (0.482 + 0.875i)10-s − 1.09i·11-s − 0.470·13-s + (0.748 + 0.862i)14-s + (−0.960 − 0.277i)16-s − 1.60·17-s + 1.17·19-s + (−0.344 + 0.938i)20-s + (0.824 − 0.716i)22-s + 1.39i·23-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((0.249−0.968i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+1/2) L(s)=((0.249−0.968i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: 0.249−0.968i0.249 - 0.968i
Analytic conductor: 2.874612.87461
Root analytic conductor: 1.695461.69546
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ360(179,⋅)\chi_{360} (179, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :1/2), 0.249−0.968i)(2,\ 360,\ (\ :1/2),\ 0.249 - 0.968i)

Particular Values

L(1)L(1) ≈\approx 1.73670+1.34575i1.73670 + 1.34575i
L(12)L(\frac12) ≈\approx 1.73670+1.34575i1.73670 + 1.34575i
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+(−0.927−1.06i)T 1 + (-0.927 - 1.06i)T
3 1 1
5 1+(−2.18−0.468i)T 1 + (-2.18 - 0.468i)T
good7 1−3.02T+7T2 1 - 3.02T + 7T^{2}
11 1+3.62iT−11T2 1 + 3.62iT - 11T^{2}
13 1+1.69T+13T2 1 + 1.69T + 13T^{2}
17 1+6.60T+17T2 1 + 6.60T + 17T^{2}
19 1−5.12T+19T2 1 - 5.12T + 19T^{2}
23 1−6.67iT−23T2 1 - 6.67iT - 23T^{2}
29 1+6.82T+29T2 1 + 6.82T + 29T^{2}
31 1+1.73iT−31T2 1 + 1.73iT - 31T^{2}
37 1+0.371T+37T2 1 + 0.371T + 37T^{2}
41 1+5.83iT−41T2 1 + 5.83iT - 41T^{2}
43 1+5.24iT−43T2 1 + 5.24iT - 43T^{2}
47 1−0.525iT−47T2 1 - 0.525iT - 47T^{2}
53 1+10.0iT−53T2 1 + 10.0iT - 53T^{2}
59 1+4.86iT−59T2 1 + 4.86iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1+13.4iT−67T2 1 + 13.4iT - 67T^{2}
71 1−2.45T+71T2 1 - 2.45T + 71T^{2}
73 1−14.5iT−73T2 1 - 14.5iT - 73T^{2}
79 1−14.1iT−79T2 1 - 14.1iT - 79T^{2}
83 1−5.79T+83T2 1 - 5.79T + 83T^{2}
89 1−10.2iT−89T2 1 - 10.2iT - 89T^{2}
97 1+9.33iT−97T2 1 + 9.33iT - 97T^{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

−11.46876624449463907848715103510, −11.11584101070880319080755532665, −9.549116718444608133342572657960, −8.738411610346438107959119657053, −7.71648949845135433309287528638, −6.78517604147374584039744828199, −5.61364050124735744638003224449, −5.08301756843702046475124345359, −3.61607289766875547149178244059, −2.13624341081602281442419021355, 1.62515988277951906713058806243, 2.54894514897133194855472053044, 4.50486785359579443612138864569, 4.95212896810919405113163276411, 6.16310825960318586195727224147, 7.30711110780857846551093464107, 8.810712012441947776661365029202, 9.582949529990553051025455515481, 10.47513516405170671963685981375, 11.28018768877054263606671756809

Graph of the ZZ-function along the critical line