↑

Properties

Label 2-2028-1.1-c3-0-56
Degree 22
Conductor 20282028
Sign −1-1
Analytic cond. 119.655119.655
Root an. cond. 10.938710.9387
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·5-s + 4·7-s + 9·9-s − 36·11-s − 18·15-s + 66·17-s − 56·19-s − 12·21-s + 96·23-s − 89·25-s − 27·27-s + 222·29-s − 260·31-s + 108·33-s + 24·35-s + 106·37-s + 90·41-s + 44·43-s + 54·45-s − 168·47-s − 327·49-s − 198·51-s + 30·53-s − 216·55-s + 168·57-s − 348·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.536·5-s + 0.215·7-s + 1/3·9-s − 0.986·11-s − 0.309·15-s + 0.941·17-s − 0.676·19-s − 0.124·21-s + 0.870·23-s − 0.711·25-s − 0.192·27-s + 1.42·29-s − 1.50·31-s + 0.569·33-s + 0.115·35-s + 0.470·37-s + 0.342·41-s + 0.156·43-s + 0.178·45-s − 0.521·47-s − 0.953·49-s − 0.543·51-s + 0.0777·53-s − 0.529·55-s + 0.390·57-s − 0.767·59-s + ⋯

Functional equation

Λ(s)=(2028s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2028s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20282028    =    22⋅3⋅1322^{2} \cdot 3 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 119.655119.655
Root analytic conductor: 10.938710.9387
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2028, ( :3/2), −1)(2,\ 2028,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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
3 1+pT 1 + p T
13 1 1
good5 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
7 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
11 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
17 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
19 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
23 1−96T+p3T2 1 - 96 T + p^{3} T^{2}
29 1−222T+p3T2 1 - 222 T + p^{3} T^{2}
31 1+260T+p3T2 1 + 260 T + p^{3} T^{2}
37 1−106T+p3T2 1 - 106 T + p^{3} T^{2}
41 1−90T+p3T2 1 - 90 T + p^{3} T^{2}
43 1−44T+p3T2 1 - 44 T + p^{3} T^{2}
47 1+168T+p3T2 1 + 168 T + p^{3} T^{2}
53 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
59 1+348T+p3T2 1 + 348 T + p^{3} T^{2}
61 1+346T+p3T2 1 + 346 T + p^{3} T^{2}
67 1−256T+p3T2 1 - 256 T + p^{3} T^{2}
71 1−168T+p3T2 1 - 168 T + p^{3} T^{2}
73 1−814T+p3T2 1 - 814 T + p^{3} T^{2}
79 1−200T+p3T2 1 - 200 T + p^{3} T^{2}
83 1+1236T+p3T2 1 + 1236 T + p^{3} T^{2}
89 1+318T+p3T2 1 + 318 T + p^{3} T^{2}
97 1−502T+p3T2 1 - 502 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

−8.283189583478755422864879115629, −7.64880575573932426851964815984, −6.73968272181127938706529291015, −5.90215757044696744174030237577, −5.27190368082399333646173217022, −4.52805242521741617109923155827, −3.32182419238289779204721457512, −2.28913875228638681264866107143, −1.22574378422677359269011871093, 0, 1.22574378422677359269011871093, 2.28913875228638681264866107143, 3.32182419238289779204721457512, 4.52805242521741617109923155827, 5.27190368082399333646173217022, 5.90215757044696744174030237577, 6.73968272181127938706529291015, 7.64880575573932426851964815984, 8.283189583478755422864879115629

Graph of the ZZ-function along the critical line