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Properties

Label 2-40e2-1.1-c3-0-38
Degree 22
Conductor 16001600
Sign 11
Analytic cond. 94.403094.4030
Root an. cond. 9.716129.71612
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 34·7-s + 9·9-s − 16·11-s + 58·13-s + 70·17-s − 4·19-s − 204·21-s + 134·23-s + 108·27-s + 242·29-s + 100·31-s + 96·33-s − 438·37-s − 348·39-s − 138·41-s + 178·43-s − 22·47-s + 813·49-s − 420·51-s + 162·53-s + 24·57-s + 268·59-s − 250·61-s + 306·63-s + 422·67-s − 804·69-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.83·7-s + 1/3·9-s − 0.438·11-s + 1.23·13-s + 0.998·17-s − 0.0482·19-s − 2.11·21-s + 1.21·23-s + 0.769·27-s + 1.54·29-s + 0.579·31-s + 0.506·33-s − 1.94·37-s − 1.42·39-s − 0.525·41-s + 0.631·43-s − 0.0682·47-s + 2.37·49-s − 1.15·51-s + 0.419·53-s + 0.0557·57-s + 0.591·59-s − 0.524·61-s + 0.611·63-s + 0.769·67-s − 1.40·69-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 11
Analytic conductor: 94.403094.4030
Root analytic conductor: 9.716129.71612
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1600, ( :3/2), 1)(2,\ 1600,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.0845500042.084550004
L(12)L(\frac12) ≈\approx 2.0845500042.084550004
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
5 1 1
good3 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
7 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
11 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
13 1−58T+p3T2 1 - 58 T + p^{3} T^{2}
17 1−70T+p3T2 1 - 70 T + p^{3} T^{2}
19 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
23 1−134T+p3T2 1 - 134 T + p^{3} T^{2}
29 1−242T+p3T2 1 - 242 T + p^{3} T^{2}
31 1−100T+p3T2 1 - 100 T + p^{3} T^{2}
37 1+438T+p3T2 1 + 438 T + p^{3} T^{2}
41 1+138T+p3T2 1 + 138 T + p^{3} T^{2}
43 1−178T+p3T2 1 - 178 T + p^{3} T^{2}
47 1+22T+p3T2 1 + 22 T + p^{3} T^{2}
53 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
59 1−268T+p3T2 1 - 268 T + p^{3} T^{2}
61 1+250T+p3T2 1 + 250 T + p^{3} T^{2}
67 1−422T+p3T2 1 - 422 T + p^{3} T^{2}
71 1+12pT+p3T2 1 + 12 p T + p^{3} T^{2}
73 1+306T+p3T2 1 + 306 T + p^{3} T^{2}
79 1+456T+p3T2 1 + 456 T + p^{3} T^{2}
83 1−434T+p3T2 1 - 434 T + p^{3} T^{2}
89 1+726T+p3T2 1 + 726 T + p^{3} T^{2}
97 1+1378T+p3T2 1 + 1378 T + p^{3} 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

−8.690163654245099286764808661693, −8.399117273267493782312454707317, −7.41818917122993891086839916674, −6.52659294551341748668611873978, −5.52351539008010257119369022222, −5.14018053822269905930552559320, −4.31080484274574571620360403811, −2.99015514151024163551326218356, −1.51415405041624097172172590820, −0.831753376812375317009992180702, 0.831753376812375317009992180702, 1.51415405041624097172172590820, 2.99015514151024163551326218356, 4.31080484274574571620360403811, 5.14018053822269905930552559320, 5.52351539008010257119369022222, 6.52659294551341748668611873978, 7.41818917122993891086839916674, 8.399117273267493782312454707317, 8.690163654245099286764808661693

Graph of the ZZ-function along the critical line