↑

Properties

Label 2-800-8.5-c5-0-47
Degree 22
Conductor 800800
Sign 0.929−0.368i0.929 - 0.368i
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  + 10.4i·3-s − 66.0·7-s + 134.·9-s + 141. i·11-s − 246. i·13-s + 297.·17-s − 174. i·19-s − 688. i·21-s − 1.57e3·23-s + 3.93e3i·27-s + 922. i·29-s + 6.19e3·31-s − 1.48e3·33-s − 1.39e4i·37-s + 2.57e3·39-s + ⋯
L(s)  = 1  + 0.669i·3-s − 0.509·7-s + 0.551·9-s + 0.353i·11-s − 0.405i·13-s + 0.249·17-s − 0.110i·19-s − 0.340i·21-s − 0.621·23-s + 1.03i·27-s + 0.203i·29-s + 1.15·31-s − 0.236·33-s − 1.67i·37-s + 0.271·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.929−0.368i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+5/2) L(s)=((0.929−0.368i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.929−0.368i0.929 - 0.368i
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), 0.929−0.368i)(2,\ 800,\ (\ :5/2),\ 0.929 - 0.368i)

Particular Values

L(3)L(3) ≈\approx 2.0614040212.061404021
L(12)L(\frac12) ≈\approx 2.0614040212.061404021
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 1−10.4iT−243T2 1 - 10.4iT - 243T^{2}
7 1+66.0T+1.68e4T2 1 + 66.0T + 1.68e4T^{2}
11 1−141.iT−1.61e5T2 1 - 141. iT - 1.61e5T^{2}
13 1+246.iT−3.71e5T2 1 + 246. iT - 3.71e5T^{2}
17 1−297.T+1.41e6T2 1 - 297.T + 1.41e6T^{2}
19 1+174.iT−2.47e6T2 1 + 174. iT - 2.47e6T^{2}
23 1+1.57e3T+6.43e6T2 1 + 1.57e3T + 6.43e6T^{2}
29 1−922.iT−2.05e7T2 1 - 922. iT - 2.05e7T^{2}
31 1−6.19e3T+2.86e7T2 1 - 6.19e3T + 2.86e7T^{2}
37 1+1.39e4iT−6.93e7T2 1 + 1.39e4iT - 6.93e7T^{2}
41 1+3.00e3T+1.15e8T2 1 + 3.00e3T + 1.15e8T^{2}
43 1+1.62e4iT−1.47e8T2 1 + 1.62e4iT - 1.47e8T^{2}
47 1−1.00e3T+2.29e8T2 1 - 1.00e3T + 2.29e8T^{2}
53 1−2.29e4iT−4.18e8T2 1 - 2.29e4iT - 4.18e8T^{2}
59 1−3.18e4iT−7.14e8T2 1 - 3.18e4iT - 7.14e8T^{2}
61 1+2.13e4iT−8.44e8T2 1 + 2.13e4iT - 8.44e8T^{2}
67 1+3.86e4iT−1.35e9T2 1 + 3.86e4iT - 1.35e9T^{2}
71 1−3.09e4T+1.80e9T2 1 - 3.09e4T + 1.80e9T^{2}
73 1−7.95e4T+2.07e9T2 1 - 7.95e4T + 2.07e9T^{2}
79 1+2.33e4T+3.07e9T2 1 + 2.33e4T + 3.07e9T^{2}
83 1+6.67e4iT−3.93e9T2 1 + 6.67e4iT - 3.93e9T^{2}
89 1−6.61e4T+5.58e9T2 1 - 6.61e4T + 5.58e9T^{2}
97 1−1.25e4T+8.58e9T2 1 - 1.25e4T + 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

−9.679241686023298532512793844410, −8.918444307312892919303298818817, −7.82381031681308660399881988319, −7.00608283402982099693872866862, −6.01226987499762360570698170975, −5.01776695996754846788992161120, −4.11516027679407723827275585769, −3.26741638218319423441235663517, −2.00904764205824002901070685715, −0.61569480339858706485956968350, 0.71253368852011956101927937950, 1.70531085800221983416647956176, 2.86827947571865473398636421542, 3.96970796570595400676776765688, 5.01503260745124788954807165700, 6.33719263950396371190328081128, 6.65646778595259152699069746984, 7.81867912056846020891718741817, 8.391260525561573212425340425987, 9.685640067457867980143483927578

Graph of the ZZ-function along the critical line