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Properties

Label 2-221952-1.1-c1-0-33
Degree 22
Conductor 221952221952
Sign −1-1
Analytic cond. 1772.291772.29
Root an. cond. 42.098642.0986
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s − 2·7-s + 9-s − 4·11-s − 2·15-s + 8·19-s + 2·21-s + 6·23-s − 25-s − 27-s + 10·29-s + 2·31-s + 4·33-s − 4·35-s + 6·37-s + 6·41-s − 12·43-s + 2·45-s + 4·47-s − 3·49-s − 4·53-s − 8·55-s − 8·57-s + 14·61-s − 2·63-s + 8·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.516·15-s + 1.83·19-s + 0.436·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s + 1.85·29-s + 0.359·31-s + 0.696·33-s − 0.676·35-s + 0.986·37-s + 0.937·41-s − 1.82·43-s + 0.298·45-s + 0.583·47-s − 3/7·49-s − 0.549·53-s − 1.07·55-s − 1.05·57-s + 1.79·61-s − 0.251·63-s + 0.977·67-s + ⋯

Functional equation

Λ(s)=(221952s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 221952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(221952s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 221952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 221952221952    =    28⋅3⋅1722^{8} \cdot 3 \cdot 17^{2}
Sign: −1-1
Analytic conductor: 1772.291772.29
Root analytic conductor: 42.098642.0986
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 221952, ( :1/2), −1)(2,\ 221952,\ (\ :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 1
3 1+T 1 + T
17 1 1
good5 1−2T+pT2 1 - 2 T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+pT2 1 + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−10T+pT2 1 - 10 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−2T+pT2 1 - 2 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1+6T+pT2 1 + 6 T + p T^{2}
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   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

−13.12358043453516, −12.80494237511053, −12.41812140179758, −11.60581402806995, −11.48921232429410, −10.83695122840328, −10.20327717568087, −9.971954505525381, −9.671413526011552, −9.139165220684915, −8.459587540374963, −7.982138808203020, −7.403099475892684, −6.931755530494956, −6.419601555051719, −6.004738958105661, −5.448112618335461, −4.992243049349001, −4.764203949685160, −3.777986059403075, −3.187936732916810, −2.693056309100028, −2.296898816044761, −1.216723928289328, −0.9236881777875847, 0, 0.9236881777875847, 1.216723928289328, 2.296898816044761, 2.693056309100028, 3.187936732916810, 3.777986059403075, 4.764203949685160, 4.992243049349001, 5.448112618335461, 6.004738958105661, 6.419601555051719, 6.931755530494956, 7.403099475892684, 7.982138808203020, 8.459587540374963, 9.139165220684915, 9.671413526011552, 9.971954505525381, 10.20327717568087, 10.83695122840328, 11.48921232429410, 11.60581402806995, 12.41812140179758, 12.80494237511053, 13.12358043453516

Graph of the ZZ-function along the critical line