↑

Properties

Label 2-72-8.5-c1-0-2
Degree 22
Conductor 7272
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 0.5749220.574922
Root an. cond. 0.7582360.758236
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 − i)2-s − 2i·4-s + 2i·5-s − 2·7-s + (−2 − 2i)8-s + (2 + 2i)10-s + 4i·13-s + (−2 + 2i)14-s − 4·16-s + 2·17-s − 4i·19-s + 4·20-s − 4·23-s + 25-s + (4 + 4i)26-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s + 0.894i·5-s − 0.755·7-s + (−0.707 − 0.707i)8-s + (0.632 + 0.632i)10-s + 1.10i·13-s + (−0.534 + 0.534i)14-s − 16-s + 0.485·17-s − 0.917i·19-s + 0.894·20-s − 0.834·23-s + 0.200·25-s + (0.784 + 0.784i)26-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+1/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 0.5749220.574922
Root analytic conductor: 0.7582360.758236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ72(37,⋅)\chi_{72} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :1/2), 0.707+0.707i)(2,\ 72,\ (\ :1/2),\ 0.707 + 0.707i)

Particular Values

L(1)L(1) ≈\approx 1.09661−0.454231i1.09661 - 0.454231i
L(12)L(\frac12) ≈\approx 1.09661−0.454231i1.09661 - 0.454231i
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+i)T 1 + (-1 + i)T
3 1 1
good5 1−2iT−5T2 1 - 2iT - 5T^{2}
7 1+2T+7T2 1 + 2T + 7T^{2}
11 1−11T2 1 - 11T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+6iT−29T2 1 + 6iT - 29T^{2}
31 1−2T+31T2 1 - 2T + 31T^{2}
37 1+8iT−37T2 1 + 8iT - 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1−12T+47T2 1 - 12T + 47T^{2}
53 1−6iT−53T2 1 - 6iT - 53T^{2}
59 1−4iT−59T2 1 - 4iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1−12iT−67T2 1 - 12iT - 67T^{2}
71 1+12T+71T2 1 + 12T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1−10T+79T2 1 - 10T + 79T^{2}
83 1−16iT−83T2 1 - 16iT - 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+2T+97T2 1 + 2T + 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

−14.27967789830314087067313413679, −13.50751769492664210204676631508, −12.27510916837260584493795738950, −11.29820456022791443126402878567, −10.24089300325458465964867268631, −9.232706728984305294883789634900, −7.04757015115889787904429590464, −5.99689698164512799731262664257, −4.13547310437713837033869561692, −2.65049428307336927197666154875, 3.42160512196739537031611417666, 5.05733945414218989480480092363, 6.19224423072169904665005262741, 7.73033600871715252944656625446, 8.780433665194772995003156464804, 10.22129357345684336042357840687, 12.08350832093766156717979241911, 12.70063404200723784944113138791, 13.63584948772805157440499549654, 14.82928336017101520461775056403

Graph of the ZZ-function along the critical line