Properties

Label 2-24e2-1.1-c3-0-26
Degree 22
Conductor 576576
Sign 1-1
Analytic cond. 33.985133.9851
Root an. cond. 5.829675.82967
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  + 10·5-s − 4·7-s + 20·11-s − 70·13-s − 90·17-s − 140·19-s + 192·23-s − 25·25-s − 134·29-s + 100·31-s − 40·35-s + 170·37-s + 110·41-s − 532·43-s + 56·47-s − 327·49-s − 430·53-s + 200·55-s − 20·59-s − 270·61-s − 700·65-s + 524·67-s + 80·71-s + 330·73-s − 80·77-s + 1.06e3·79-s − 1.18e3·83-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.215·7-s + 0.548·11-s − 1.49·13-s − 1.28·17-s − 1.69·19-s + 1.74·23-s − 1/5·25-s − 0.858·29-s + 0.579·31-s − 0.193·35-s + 0.755·37-s + 0.419·41-s − 1.88·43-s + 0.173·47-s − 0.953·49-s − 1.11·53-s + 0.490·55-s − 0.0441·59-s − 0.566·61-s − 1.33·65-s + 0.955·67-s + 0.133·71-s + 0.529·73-s − 0.118·77-s + 1.50·79-s − 1.57·83-s + ⋯

Functional equation

Λ(s)=(576s/2ΓC(s)L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(576s/2ΓC(s+3/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 576576    =    26322^{6} \cdot 3^{2}
Sign: 1-1
Analytic conductor: 33.985133.9851
Root analytic conductor: 5.829675.82967
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 576, ( :3/2), 1)(2,\ 576,\ (\ :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(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
good5 12pT+p3T2 1 - 2 p T + p^{3} T^{2}
7 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
11 120T+p3T2 1 - 20 T + p^{3} T^{2}
13 1+70T+p3T2 1 + 70 T + p^{3} T^{2}
17 1+90T+p3T2 1 + 90 T + p^{3} T^{2}
19 1+140T+p3T2 1 + 140 T + p^{3} T^{2}
23 1192T+p3T2 1 - 192 T + p^{3} T^{2}
29 1+134T+p3T2 1 + 134 T + p^{3} T^{2}
31 1100T+p3T2 1 - 100 T + p^{3} T^{2}
37 1170T+p3T2 1 - 170 T + p^{3} T^{2}
41 1110T+p3T2 1 - 110 T + p^{3} T^{2}
43 1+532T+p3T2 1 + 532 T + p^{3} T^{2}
47 156T+p3T2 1 - 56 T + p^{3} T^{2}
53 1+430T+p3T2 1 + 430 T + p^{3} T^{2}
59 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
61 1+270T+p3T2 1 + 270 T + p^{3} T^{2}
67 1524T+p3T2 1 - 524 T + p^{3} T^{2}
71 180T+p3T2 1 - 80 T + p^{3} T^{2}
73 1330T+p3T2 1 - 330 T + p^{3} T^{2}
79 11060T+p3T2 1 - 1060 T + p^{3} T^{2}
83 1+1188T+p3T2 1 + 1188 T + p^{3} T^{2}
89 1+1274T+p3T2 1 + 1274 T + p^{3} T^{2}
97 1+590T+p3T2 1 + 590 T + p^{3} T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.694397508797212442745035212872, −9.245377962400217435059977358078, −8.208332681649884755425609288332, −6.89199337660583853317715284780, −6.41716780902006901716200848148, −5.16741838372930162984051879029, −4.30632769635815019718880272138, −2.74634064417202034913997095541, −1.81110578588999287003461155390, 0, 1.81110578588999287003461155390, 2.74634064417202034913997095541, 4.30632769635815019718880272138, 5.16741838372930162984051879029, 6.41716780902006901716200848148, 6.89199337660583853317715284780, 8.208332681649884755425609288332, 9.245377962400217435059977358078, 9.694397508797212442745035212872

Graph of the ZZ-function along the critical line