↑

Properties

Label 2-210-1.1-c5-0-19
Degree 22
Conductor 210210
Sign −1-1
Analytic cond. 33.680633.6806
Root an. cond. 5.803495.80349
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 9·3-s + 16·4-s − 25·5-s + 36·6-s − 49·7-s + 64·8-s + 81·9-s − 100·10-s − 668·11-s + 144·12-s − 366·13-s − 196·14-s − 225·15-s + 256·16-s − 34·17-s + 324·18-s − 2.01e3·19-s − 400·20-s − 441·21-s − 2.67e3·22-s + 1.90e3·23-s + 576·24-s + 625·25-s − 1.46e3·26-s + 729·27-s − 784·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.66·11-s + 0.288·12-s − 0.600·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.0285·17-s + 0.235·18-s − 1.28·19-s − 0.223·20-s − 0.218·21-s − 1.17·22-s + 0.752·23-s + 0.204·24-s + 1/5·25-s − 0.424·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(210s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(210s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 210210    =    2⋅3⋅5⋅72 \cdot 3 \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 33.680633.6806
Root analytic conductor: 5.803495.80349
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 210, ( :5/2), −1)(2,\ 210,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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−p2T 1 - p^{2} T
3 1−p2T 1 - p^{2} T
5 1+p2T 1 + p^{2} T
7 1+p2T 1 + p^{2} T
good11 1+668T+p5T2 1 + 668 T + p^{5} T^{2}
13 1+366T+p5T2 1 + 366 T + p^{5} T^{2}
17 1+2pT+p5T2 1 + 2 p T + p^{5} T^{2}
19 1+2016T+p5T2 1 + 2016 T + p^{5} T^{2}
23 1−1908T+p5T2 1 - 1908 T + p^{5} T^{2}
29 1+5754T+p5T2 1 + 5754 T + p^{5} T^{2}
31 1+52T+p5T2 1 + 52 T + p^{5} T^{2}
37 1+10594T+p5T2 1 + 10594 T + p^{5} T^{2}
41 1+418T+p5T2 1 + 418 T + p^{5} T^{2}
43 1−6676T+p5T2 1 - 6676 T + p^{5} T^{2}
47 1−1472T+p5T2 1 - 1472 T + p^{5} T^{2}
53 1−19834T+p5T2 1 - 19834 T + p^{5} T^{2}
59 1−10492T+p5T2 1 - 10492 T + p^{5} T^{2}
61 1+38810T+p5T2 1 + 38810 T + p^{5} T^{2}
67 1+61428T+p5T2 1 + 61428 T + p^{5} T^{2}
71 1+7936T+p5T2 1 + 7936 T + p^{5} T^{2}
73 1−21134T+p5T2 1 - 21134 T + p^{5} T^{2}
79 1−87088T+p5T2 1 - 87088 T + p^{5} T^{2}
83 1+1244pT+p5T2 1 + 1244 p T + p^{5} T^{2}
89 1+49490T+p5T2 1 + 49490 T + p^{5} T^{2}
97 1−125630T+p5T2 1 - 125630 T + p^{5} 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

−10.95948543027083823034815999019, −10.24742061042926004887010221875, −8.909754932660265391381729438469, −7.80938077767942718211155766737, −7.01091259173520728062816764614, −5.56542516812400273773836393562, −4.47135984739346333135235060796, −3.21530318792644663060752083770, −2.20661752654367061804889680007, 0, 2.20661752654367061804889680007, 3.21530318792644663060752083770, 4.47135984739346333135235060796, 5.56542516812400273773836393562, 7.01091259173520728062816764614, 7.80938077767942718211155766737, 8.909754932660265391381729438469, 10.24742061042926004887010221875, 10.95948543027083823034815999019

Graph of the ZZ-function along the critical line