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Properties

Label 2-40e2-1.1-c3-0-78
Degree 22
Conductor 16001600
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 6·7-s − 23·9-s + 32·11-s − 38·13-s − 26·17-s + 100·19-s − 12·21-s − 78·23-s + 100·27-s + 50·29-s + 108·31-s − 64·33-s + 266·37-s + 76·39-s + 22·41-s − 442·43-s − 514·47-s − 307·49-s + 52·51-s + 2·53-s − 200·57-s + 500·59-s + 518·61-s − 138·63-s − 126·67-s + 156·69-s + ⋯
L(s)  = 1  − 0.384·3-s + 0.323·7-s − 0.851·9-s + 0.877·11-s − 0.810·13-s − 0.370·17-s + 1.20·19-s − 0.124·21-s − 0.707·23-s + 0.712·27-s + 0.320·29-s + 0.625·31-s − 0.337·33-s + 1.18·37-s + 0.312·39-s + 0.0838·41-s − 1.56·43-s − 1.59·47-s − 0.895·49-s + 0.142·51-s + 0.00518·53-s − 0.464·57-s + 1.10·59-s + 1.08·61-s − 0.275·63-s − 0.229·67-s + 0.272·69-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: −1-1
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: 11
Selberg data: (2, 1600, ( :3/2), −1)(2,\ 1600,\ (\ :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
5 1 1
good3 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
7 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
11 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
13 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
17 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
19 1−100T+p3T2 1 - 100 T + p^{3} T^{2}
23 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
29 1−50T+p3T2 1 - 50 T + p^{3} T^{2}
31 1−108T+p3T2 1 - 108 T + p^{3} T^{2}
37 1−266T+p3T2 1 - 266 T + p^{3} T^{2}
41 1−22T+p3T2 1 - 22 T + p^{3} T^{2}
43 1+442T+p3T2 1 + 442 T + p^{3} T^{2}
47 1+514T+p3T2 1 + 514 T + p^{3} T^{2}
53 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
59 1−500T+p3T2 1 - 500 T + p^{3} T^{2}
61 1−518T+p3T2 1 - 518 T + p^{3} T^{2}
67 1+126T+p3T2 1 + 126 T + p^{3} T^{2}
71 1+412T+p3T2 1 + 412 T + p^{3} T^{2}
73 1−878T+p3T2 1 - 878 T + p^{3} T^{2}
79 1+600T+p3T2 1 + 600 T + p^{3} T^{2}
83 1+282T+p3T2 1 + 282 T + p^{3} T^{2}
89 1+150T+p3T2 1 + 150 T + p^{3} T^{2}
97 1+386T+p3T2 1 + 386 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

−8.549635747050430196113879821730, −7.959014958217588211203349273307, −6.91251799696926957634376922130, −6.24093483968302457641386762894, −5.30324480511294971580678990386, −4.61968981095360200289027508977, −3.47865249188224677109850834953, −2.47432493945510607249898334872, −1.22786431127161040378292335997, 0, 1.22786431127161040378292335997, 2.47432493945510607249898334872, 3.47865249188224677109850834953, 4.61968981095360200289027508977, 5.30324480511294971580678990386, 6.24093483968302457641386762894, 6.91251799696926957634376922130, 7.959014958217588211203349273307, 8.549635747050430196113879821730

Graph of the ZZ-function along the critical line