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Properties

Label 2-95e2-1.1-c1-0-237
Degree 22
Conductor 90259025
Sign −1-1
Analytic cond. 72.064972.0649
Root an. cond. 8.489108.48910
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  − 2·4-s − 3·7-s − 3·9-s − 5·11-s + 4·16-s + 7·17-s + 4·23-s + 6·28-s + 6·36-s + 43-s + 10·44-s − 13·47-s + 2·49-s + 15·61-s + 9·63-s − 8·64-s − 14·68-s + 11·73-s + 15·77-s + 9·81-s + 16·83-s − 8·92-s + 15·99-s + 10·101-s − 12·112-s − 21·119-s + ⋯
L(s)  = 1  − 4-s − 1.13·7-s − 9-s − 1.50·11-s + 16-s + 1.69·17-s + 0.834·23-s + 1.13·28-s + 36-s + 0.152·43-s + 1.50·44-s − 1.89·47-s + 2/7·49-s + 1.92·61-s + 1.13·63-s − 64-s − 1.69·68-s + 1.28·73-s + 1.70·77-s + 81-s + 1.75·83-s − 0.834·92-s + 1.50·99-s + 0.995·101-s − 1.13·112-s − 1.92·119-s + ⋯

Functional equation

Λ(s)=(9025s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9025s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 90259025    =    52⋅1925^{2} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 72.064972.0649
Root analytic conductor: 8.489108.48910
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9025, ( :1/2), −1)(2,\ 9025,\ (\ :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)
bad5 1 1
19 1 1
good2 1+pT2 1 + p T^{2}
3 1+pT2 1 + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−7T+pT2 1 - 7 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+pT2 1 + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1+13T+pT2 1 + 13 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−15T+pT2 1 - 15 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+pT2 1 + 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

−7.61824713974920993594504764555, −6.63088607414418398663848705247, −5.84015582650017493723925533798, −5.29579516351674176013179187813, −4.85335432134469239130548931207, −3.48854845472212746808461755996, −3.31297016434738025974202284029, −2.43814410977130447009590932247, −0.871520748833572046051686868602, 0, 0.871520748833572046051686868602, 2.43814410977130447009590932247, 3.31297016434738025974202284029, 3.48854845472212746808461755996, 4.85335432134469239130548931207, 5.29579516351674176013179187813, 5.84015582650017493723925533798, 6.63088607414418398663848705247, 7.61824713974920993594504764555

Graph of the ZZ-function along the critical line