↑

Properties

Label 2-5760-8.5-c1-0-68
Degree 22
Conductor 57605760
Sign −0.707+0.707i-0.707 + 0.707i
Analytic cond. 45.993845.9938
Root an. cond. 6.781876.78187
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  + i·5-s + 2·7-s − 2i·11-s + 2i·13-s − 4·17-s − 4i·19-s − 4·23-s − 25-s − 2i·29-s + 4·31-s + 2i·35-s + 2i·37-s − 6·41-s − 4i·43-s − 8·47-s + ⋯
L(s)  = 1  + 0.447i·5-s + 0.755·7-s − 0.603i·11-s + 0.554i·13-s − 0.970·17-s − 0.917i·19-s − 0.834·23-s − 0.200·25-s − 0.371i·29-s + 0.718·31-s + 0.338i·35-s + 0.328i·37-s − 0.937·41-s − 0.609i·43-s − 1.16·47-s + ⋯

Functional equation

Λ(s)=(5760s/2 ΓC(s) L(s)=((−0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5760s/2 ΓC(s+1/2) L(s)=((−0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{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: 57605760    =    27⋅32⋅52^{7} \cdot 3^{2} \cdot 5
Sign: −0.707+0.707i-0.707 + 0.707i
Analytic conductor: 45.993845.9938
Root analytic conductor: 6.781876.78187
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5760(2881,⋅)\chi_{5760} (2881, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5760, ( :1/2), −0.707+0.707i)(2,\ 5760,\ (\ :1/2),\ -0.707 + 0.707i)

Particular Values

L(1)L(1) ≈\approx 0.64412973100.6441297310
L(12)L(\frac12) ≈\approx 0.64412973100.6441297310
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 1
5 1−iT 1 - iT
good7 1−2T+7T2 1 - 2T + 7T^{2}
11 1+2iT−11T2 1 + 2iT - 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
17 1+4T+17T2 1 + 4T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+2iT−29T2 1 + 2iT - 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1+10iT−53T2 1 + 10iT - 53T^{2}
59 1−6iT−59T2 1 - 6iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1+12iT−67T2 1 + 12iT - 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−16iT−83T2 1 - 16iT - 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+14T+97T2 1 + 14T + 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

−8.082400256523640018282581260327, −6.93142814549780539705180739273, −6.62110122072285913163900606985, −5.72619023313681491768728941686, −4.85240049707536555008275018235, −4.26960590191886573809103148557, −3.32090664175627645551676191427, −2.40616837131909052644371411417, −1.59956901693486321499320967536, −0.15633567195328539711829573712, 1.33931723470777923243463564057, 2.05179052624208101002207894400, 3.12591327207140560339997469979, 4.19103009978474118572067126364, 4.67909559982567454125530140772, 5.45883953832016902731987089983, 6.20615840572147053438454479915, 7.00292078738893923101586874100, 7.898336721004458663399858952717, 8.214237187798307473687710618935

Graph of the ZZ-function along the critical line