↑

Properties

Label 2-800-8.5-c5-0-61
Degree 22
Conductor 800800
Sign 0.916−0.401i0.916 - 0.401i
Analytic cond. 128.307128.307
Root an. cond. 11.327211.3272
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 10.8i·3-s + 163.·7-s + 125.·9-s − 321. i·11-s − 128. i·13-s + 2.11e3·17-s + 1.45e3i·19-s + 1.77e3i·21-s − 1.23e3·23-s + 3.99e3i·27-s + 4.07e3i·29-s + 3.95e3·31-s + 3.48e3·33-s − 1.06e4i·37-s + 1.38e3·39-s + ⋯
L(s)  = 1  + 0.694i·3-s + 1.26·7-s + 0.517·9-s − 0.801i·11-s − 0.210i·13-s + 1.77·17-s + 0.924i·19-s + 0.876i·21-s − 0.485·23-s + 1.05i·27-s + 0.899i·29-s + 0.739·31-s + 0.556·33-s − 1.27i·37-s + 0.146·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.916−0.401i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.401i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+5/2) L(s)=((0.916−0.401i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.916 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.916−0.401i0.916 - 0.401i
Analytic conductor: 128.307128.307
Root analytic conductor: 11.327211.3272
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ800(401,⋅)\chi_{800} (401, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :5/2), 0.916−0.401i)(2,\ 800,\ (\ :5/2),\ 0.916 - 0.401i)

Particular Values

L(3)L(3) ≈\approx 3.2760705543.276070554
L(12)L(\frac12) ≈\approx 3.2760705543.276070554
L(72)L(\frac{7}{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
5 1 1
good3 1−10.8iT−243T2 1 - 10.8iT - 243T^{2}
7 1−163.T+1.68e4T2 1 - 163.T + 1.68e4T^{2}
11 1+321.iT−1.61e5T2 1 + 321. iT - 1.61e5T^{2}
13 1+128.iT−3.71e5T2 1 + 128. iT - 3.71e5T^{2}
17 1−2.11e3T+1.41e6T2 1 - 2.11e3T + 1.41e6T^{2}
19 1−1.45e3iT−2.47e6T2 1 - 1.45e3iT - 2.47e6T^{2}
23 1+1.23e3T+6.43e6T2 1 + 1.23e3T + 6.43e6T^{2}
29 1−4.07e3iT−2.05e7T2 1 - 4.07e3iT - 2.05e7T^{2}
31 1−3.95e3T+2.86e7T2 1 - 3.95e3T + 2.86e7T^{2}
37 1+1.06e4iT−6.93e7T2 1 + 1.06e4iT - 6.93e7T^{2}
41 1+5.90e3T+1.15e8T2 1 + 5.90e3T + 1.15e8T^{2}
43 1+1.64e4iT−1.47e8T2 1 + 1.64e4iT - 1.47e8T^{2}
47 1−2.32e4T+2.29e8T2 1 - 2.32e4T + 2.29e8T^{2}
53 1+3.06e4iT−4.18e8T2 1 + 3.06e4iT - 4.18e8T^{2}
59 1−2.52e4iT−7.14e8T2 1 - 2.52e4iT - 7.14e8T^{2}
61 1+3.91e4iT−8.44e8T2 1 + 3.91e4iT - 8.44e8T^{2}
67 1−2.08e4iT−1.35e9T2 1 - 2.08e4iT - 1.35e9T^{2}
71 1+1.38e4T+1.80e9T2 1 + 1.38e4T + 1.80e9T^{2}
73 1−4.34e4T+2.07e9T2 1 - 4.34e4T + 2.07e9T^{2}
79 1−1.25e4T+3.07e9T2 1 - 1.25e4T + 3.07e9T^{2}
83 1+6.68e3iT−3.93e9T2 1 + 6.68e3iT - 3.93e9T^{2}
89 1+9.04e4T+5.58e9T2 1 + 9.04e4T + 5.58e9T^{2}
97 1+1.49e5T+8.58e9T2 1 + 1.49e5T + 8.58e9T^{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

−9.739353086142610263781823420249, −8.624613288568801105449537514887, −7.986923941088818240954468357385, −7.16456454288364769138357897848, −5.70530996476060679249241708415, −5.21485844081377832800335016336, −4.08633142180622286205486359532, −3.35732709908559186018502231512, −1.82297973055510487792006201151, −0.861632574432560896443446796816, 0.943379521777699350879158837354, 1.62023291241129660448397992529, 2.69114412503748785940609290897, 4.22823348695818886718371934709, 4.90412359069932237208142370711, 6.01981283687711223955347470201, 7.06732554340907879891989474873, 7.74533402276687495466015300161, 8.287628451166840968394585625928, 9.585189552469009235325227743378

Graph of the ZZ-function along the critical line