↑

Properties

Label 2-200-1.1-c3-0-4
Degree 22
Conductor 200200
Sign 11
Analytic cond. 11.800311.8003
Root an. cond. 3.435163.43516
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  + 5·3-s + 2·7-s − 2·9-s + 39·11-s + 84·13-s − 61·17-s + 151·19-s + 10·21-s − 58·23-s − 145·27-s + 192·29-s − 18·31-s + 195·33-s − 138·37-s + 420·39-s + 229·41-s − 164·43-s − 212·47-s − 339·49-s − 305·51-s + 578·53-s + 755·57-s − 336·59-s + 858·61-s − 4·63-s − 209·67-s − 290·69-s + ⋯
L(s)  = 1  + 0.962·3-s + 0.107·7-s − 0.0740·9-s + 1.06·11-s + 1.79·13-s − 0.870·17-s + 1.82·19-s + 0.103·21-s − 0.525·23-s − 1.03·27-s + 1.22·29-s − 0.104·31-s + 1.02·33-s − 0.613·37-s + 1.72·39-s + 0.872·41-s − 0.581·43-s − 0.657·47-s − 0.988·49-s − 0.837·51-s + 1.49·53-s + 1.75·57-s − 0.741·59-s + 1.80·61-s − 0.00799·63-s − 0.381·67-s − 0.505·69-s + ⋯

Functional equation

Λ(s)=(200s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(200s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23⋅522^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 11.800311.8003
Root analytic conductor: 3.435163.43516
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 200, ( :3/2), 1)(2,\ 200,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.5956731222.595673122
L(12)L(\frac12) ≈\approx 2.5956731222.595673122
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 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
7 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
11 1−39T+p3T2 1 - 39 T + p^{3} T^{2}
13 1−84T+p3T2 1 - 84 T + p^{3} T^{2}
17 1+61T+p3T2 1 + 61 T + p^{3} T^{2}
19 1−151T+p3T2 1 - 151 T + p^{3} T^{2}
23 1+58T+p3T2 1 + 58 T + p^{3} T^{2}
29 1−192T+p3T2 1 - 192 T + p^{3} T^{2}
31 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
37 1+138T+p3T2 1 + 138 T + p^{3} T^{2}
41 1−229T+p3T2 1 - 229 T + p^{3} T^{2}
43 1+164T+p3T2 1 + 164 T + p^{3} T^{2}
47 1+212T+p3T2 1 + 212 T + p^{3} T^{2}
53 1−578T+p3T2 1 - 578 T + p^{3} T^{2}
59 1+336T+p3T2 1 + 336 T + p^{3} T^{2}
61 1−858T+p3T2 1 - 858 T + p^{3} T^{2}
67 1+209T+p3T2 1 + 209 T + p^{3} T^{2}
71 1+780T+p3T2 1 + 780 T + p^{3} T^{2}
73 1+403T+p3T2 1 + 403 T + p^{3} T^{2}
79 1+230T+p3T2 1 + 230 T + p^{3} T^{2}
83 1+1293T+p3T2 1 + 1293 T + p^{3} T^{2}
89 1+1369T+p3T2 1 + 1369 T + p^{3} T^{2}
97 1−382T+p3T2 1 - 382 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

−11.85340200077995382974986278357, −11.18557493684478897875068215159, −9.807469515722484982055208540286, −8.842760142606351954510600134901, −8.277835662904540116112287939789, −6.92577140210464799514359797394, −5.78253706349857504577230947665, −4.07965010062562905854019559774, −3.08039223659535500209288017213, −1.39244343621259215093755400948, 1.39244343621259215093755400948, 3.08039223659535500209288017213, 4.07965010062562905854019559774, 5.78253706349857504577230947665, 6.92577140210464799514359797394, 8.277835662904540116112287939789, 8.842760142606351954510600134901, 9.807469515722484982055208540286, 11.18557493684478897875068215159, 11.85340200077995382974986278357

Graph of the ZZ-function along the critical line