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Properties

Label 2-208725-1.1-c1-0-43
Degree 22
Conductor 208725208725
Sign −1-1
Analytic cond. 1666.671666.67
Root an. cond. 40.824940.8249
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  + 2-s + 3-s − 4-s + 6-s − 2·7-s − 3·8-s + 9-s − 12-s − 2·13-s − 2·14-s − 16-s + 18-s + 4·19-s − 2·21-s + 23-s − 3·24-s − 2·26-s + 27-s + 2·28-s − 4·29-s + 5·32-s − 36-s − 2·37-s + 4·38-s − 2·39-s − 4·41-s − 2·42-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.436·21-s + 0.208·23-s − 0.612·24-s − 0.392·26-s + 0.192·27-s + 0.377·28-s − 0.742·29-s + 0.883·32-s − 1/6·36-s − 0.328·37-s + 0.648·38-s − 0.320·39-s − 0.624·41-s − 0.308·42-s + ⋯

Functional equation

Λ(s)=(208725s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(208725s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 208725208725    =    3⋅52⋅112⋅233 \cdot 5^{2} \cdot 11^{2} \cdot 23
Sign: −1-1
Analytic conductor: 1666.671666.67
Root analytic conductor: 40.824940.8249
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 208725, ( :1/2), −1)(2,\ 208725,\ (\ :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)
bad3 1−T 1 - T
5 1 1
11 1 1
23 1−T 1 - T
good2 1−T+pT2 1 - T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+4T+pT2 1 + 4 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + 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+18T+pT2 1 + 18 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−13.28340808635157, −12.93540059287706, −12.41715591377661, −12.08151884427378, −11.49862268818983, −11.07813176357095, −10.12912846308737, −9.903045721427726, −9.631916978567122, −8.969695952954687, −8.580327782059112, −8.181416092087840, −7.397778809128558, −7.009995045785625, −6.610923068066844, −5.782631645342593, −5.469196138923764, −4.993924579477822, −4.358694690911181, −3.767721317287763, −3.483444743590081, −2.845364559897522, −2.437293741928117, −1.582216133737111, −0.7435283500706543, 0, 0.7435283500706543, 1.582216133737111, 2.437293741928117, 2.845364559897522, 3.483444743590081, 3.767721317287763, 4.358694690911181, 4.993924579477822, 5.469196138923764, 5.782631645342593, 6.610923068066844, 7.009995045785625, 7.397778809128558, 8.181416092087840, 8.580327782059112, 8.969695952954687, 9.631916978567122, 9.903045721427726, 10.12912846308737, 11.07813176357095, 11.49862268818983, 12.08151884427378, 12.41715591377661, 12.93540059287706, 13.28340808635157

Graph of the ZZ-function along the critical line