↑

Properties

Label 2-800-1.1-c3-0-35
Degree 22
Conductor 800800
Sign 11
Analytic cond. 47.201547.2015
Root an. cond. 6.870336.87033
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s + 16·7-s + 37·9-s + 40·11-s + 50·13-s + 30·17-s − 40·19-s + 128·21-s + 48·23-s + 80·27-s − 34·29-s − 320·31-s + 320·33-s − 310·37-s + 400·39-s + 410·41-s + 152·43-s − 416·47-s − 87·49-s + 240·51-s + 410·53-s − 320·57-s + 200·59-s + 30·61-s + 592·63-s + 776·67-s + 384·69-s + ⋯
L(s)  = 1  + 1.53·3-s + 0.863·7-s + 1.37·9-s + 1.09·11-s + 1.06·13-s + 0.428·17-s − 0.482·19-s + 1.33·21-s + 0.435·23-s + 0.570·27-s − 0.217·29-s − 1.85·31-s + 1.68·33-s − 1.37·37-s + 1.64·39-s + 1.56·41-s + 0.539·43-s − 1.29·47-s − 0.253·49-s + 0.658·51-s + 1.06·53-s − 0.743·57-s + 0.441·59-s + 0.0629·61-s + 1.18·63-s + 1.41·67-s + 0.669·69-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 11
Analytic conductor: 47.201547.2015
Root analytic conductor: 6.870336.87033
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 800, ( :3/2), 1)(2,\ 800,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 4.5725271004.572527100
L(12)L(\frac12) ≈\approx 4.5725271004.572527100
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
5 1 1
good3 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
7 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
11 1−40T+p3T2 1 - 40 T + p^{3} T^{2}
13 1−50T+p3T2 1 - 50 T + p^{3} T^{2}
17 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
19 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
23 1−48T+p3T2 1 - 48 T + p^{3} T^{2}
29 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
31 1+320T+p3T2 1 + 320 T + p^{3} T^{2}
37 1+310T+p3T2 1 + 310 T + p^{3} T^{2}
41 1−10pT+p3T2 1 - 10 p T + p^{3} T^{2}
43 1−152T+p3T2 1 - 152 T + p^{3} T^{2}
47 1+416T+p3T2 1 + 416 T + p^{3} T^{2}
53 1−410T+p3T2 1 - 410 T + p^{3} T^{2}
59 1−200T+p3T2 1 - 200 T + p^{3} T^{2}
61 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
67 1−776T+p3T2 1 - 776 T + p^{3} T^{2}
71 1+400T+p3T2 1 + 400 T + p^{3} T^{2}
73 1−630T+p3T2 1 - 630 T + p^{3} T^{2}
79 1−1120T+p3T2 1 - 1120 T + p^{3} T^{2}
83 1−552T+p3T2 1 - 552 T + p^{3} T^{2}
89 1+326T+p3T2 1 + 326 T + p^{3} T^{2}
97 1−110T+p3T2 1 - 110 T + p^{3} 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

−9.500063986950754882251664983094, −8.946696934215502822814021631729, −8.312263393578806139327757745640, −7.55044132704013839292442313112, −6.59784340661538159549951933926, −5.34489347802331128251369776429, −4.01429101447137920022477466250, −3.50765195689488865882793651829, −2.13780006390465741062937893254, −1.28609364578193050044790091343, 1.28609364578193050044790091343, 2.13780006390465741062937893254, 3.50765195689488865882793651829, 4.01429101447137920022477466250, 5.34489347802331128251369776429, 6.59784340661538159549951933926, 7.55044132704013839292442313112, 8.312263393578806139327757745640, 8.946696934215502822814021631729, 9.500063986950754882251664983094

Graph of the ZZ-function along the critical line