↑

Properties

Label 2-162-9.4-c3-0-10
Degree 22
Conductor 162162
Sign −0.766+0.642i-0.766 + 0.642i
Analytic cond. 9.558309.55830
Root an. cond. 3.091653.09165
Motivic weight 33
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 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (10.5 − 18.1i)5-s + (−4 − 6.92i)7-s + 7.99·8-s − 42·10-s + (18 + 31.1i)11-s + (24.5 − 42.4i)13-s + (−7.99 + 13.8i)14-s + (−8 − 13.8i)16-s − 21·17-s − 112·19-s + (42 + 72.7i)20-s + (36 − 62.3i)22-s + (90 − 155. i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.939 − 1.62i)5-s + (−0.215 − 0.374i)7-s + 0.353·8-s − 1.32·10-s + (0.493 + 0.854i)11-s + (0.522 − 0.905i)13-s + (−0.152 + 0.264i)14-s + (−0.125 − 0.216i)16-s − 0.299·17-s − 1.35·19-s + (0.469 + 0.813i)20-s + (0.348 − 0.604i)22-s + (0.815 − 1.41i)23-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=((−0.766+0.642i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+3/2) L(s)=((−0.766+0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: −0.766+0.642i-0.766 + 0.642i
Analytic conductor: 9.558309.55830
Root analytic conductor: 3.091653.09165
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ162(109,⋅)\chi_{162} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 162, ( :3/2), −0.766+0.642i)(2,\ 162,\ (\ :3/2),\ -0.766 + 0.642i)

Particular Values

L(2)L(2) ≈\approx 0.479782−1.31819i0.479782 - 1.31819i
L(12)L(\frac12) ≈\approx 0.479782−1.31819i0.479782 - 1.31819i
L(52)L(\frac{5}{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+1.73i)T 1 + (1 + 1.73i)T
3 1 1
good5 1+(−10.5+18.1i)T+(−62.5−108.i)T2 1 + (-10.5 + 18.1i)T + (-62.5 - 108. i)T^{2}
7 1+(4+6.92i)T+(−171.5+297.i)T2 1 + (4 + 6.92i)T + (-171.5 + 297. i)T^{2}
11 1+(−18−31.1i)T+(−665.5+1.15e3i)T2 1 + (-18 - 31.1i)T + (-665.5 + 1.15e3i)T^{2}
13 1+(−24.5+42.4i)T+(−1.09e3−1.90e3i)T2 1 + (-24.5 + 42.4i)T + (-1.09e3 - 1.90e3i)T^{2}
17 1+21T+4.91e3T2 1 + 21T + 4.91e3T^{2}
19 1+112T+6.85e3T2 1 + 112T + 6.85e3T^{2}
23 1+(−90+155.i)T+(−6.08e3−1.05e4i)T2 1 + (-90 + 155. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(67.5+116.i)T+(−1.21e4+2.11e4i)T2 1 + (67.5 + 116. i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+(154−266.i)T+(−1.48e4−2.57e4i)T2 1 + (154 - 266. i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+T+5.06e4T2 1 + T + 5.06e4T^{2}
41 1+(21−36.3i)T+(−3.44e4−5.96e4i)T2 1 + (21 - 36.3i)T + (-3.44e4 - 5.96e4i)T^{2}
43 1+(10+17.3i)T+(−3.97e4+6.88e4i)T2 1 + (10 + 17.3i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+(−42−72.7i)T+(−5.19e4+8.99e4i)T2 1 + (-42 - 72.7i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1−174T+1.48e5T2 1 - 174T + 1.48e5T^{2}
59 1+(−252+436.i)T+(−1.02e5−1.77e5i)T2 1 + (-252 + 436. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−192.5−333.i)T+(−1.13e5+1.96e5i)T2 1 + (-192.5 - 333. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(136−235.i)T+(−1.50e5−2.60e5i)T2 1 + (136 - 235. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1−888T+3.57e5T2 1 - 888T + 3.57e5T^{2}
73 1−371T+3.89e5T2 1 - 371T + 3.89e5T^{2}
79 1+(−326−564.i)T+(−2.46e5+4.26e5i)T2 1 + (-326 - 564. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+(−42−72.7i)T+(−2.85e5+4.95e5i)T2 1 + (-42 - 72.7i)T + (-2.85e5 + 4.95e5i)T^{2}
89 1+21T+7.04e5T2 1 + 21T + 7.04e5T^{2}
97 1+(−623−1.07e3i)T+(−4.56e5+7.90e5i)T2 1 + (-623 - 1.07e3i)T + (-4.56e5 + 7.90e5i)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

−12.37365569412686385313093478142, −10.79647128807278007024727004397, −9.951693221786258885633588973802, −8.972271204568524115182737871001, −8.344339804060036266096892806714, −6.63646743232648757435686047849, −5.19645933390127768821127762017, −4.13320893001743525914837271162, −2.07105184789143414197317796633, −0.75160116229901187165317760534, 2.03371887894989969365295511358, 3.61725175184557564453483206474, 5.74627633160831347732966629430, 6.41257781912142378158782433183, 7.27378153000945299767131004090, 8.871044944957653067071524676734, 9.575941390545464730437556384030, 10.83286639728749434244102639692, 11.32444565377788754001933627804, 13.19636071814474684010336587790

Graph of the ZZ-function along the critical line