↑

Properties

Label 2-2550-1.1-c1-0-33
Degree 22
Conductor 25502550
Sign −1-1
Analytic cond. 20.361820.3618
Root an. cond. 4.512414.51241
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 4·11-s − 12-s − 2·13-s + 16-s − 17-s − 18-s − 4·19-s − 4·22-s − 4·23-s + 24-s + 2·26-s − 27-s + 2·29-s − 4·31-s − 32-s − 4·33-s + 34-s + 36-s + 6·37-s + 4·38-s + 2·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.852·22-s − 0.834·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.176·32-s − 0.696·33-s + 0.171·34-s + 1/6·36-s + 0.986·37-s + 0.648·38-s + 0.320·39-s + ⋯

Functional equation

Λ(s)=(2550s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2550s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25502550    =    2⋅3⋅52⋅172 \cdot 3 \cdot 5^{2} \cdot 17
Sign: −1-1
Analytic conductor: 20.361820.3618
Root analytic conductor: 4.512414.51241
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2550, ( :1/2), −1)(2,\ 2550,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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+T 1 + T
3 1+T 1 + T
5 1 1
17 1+T 1 + T
good7 1+pT2 1 + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p 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

−8.508076357926522304587707936931, −7.85391852680015313417178593889, −6.81293701478766628611331020079, −6.47749789975439920149959740273, −5.55607683757389846715119248044, −4.51444126045046681585118943777, −3.71442293834107121729389823605, −2.36951231460413935519804036920, −1.37551986425423722963512172703, 0, 1.37551986425423722963512172703, 2.36951231460413935519804036920, 3.71442293834107121729389823605, 4.51444126045046681585118943777, 5.55607683757389846715119248044, 6.47749789975439920149959740273, 6.81293701478766628611331020079, 7.85391852680015313417178593889, 8.508076357926522304587707936931

Graph of the ZZ-function along the critical line