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Properties

Label 2-39e2-1.1-c3-0-102
Degree 22
Conductor 15211521
Sign −1-1
Analytic cond. 89.741989.7419
Root an. cond. 9.473229.47322
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  − 5·2-s + 17·4-s − 7·5-s + 13·7-s − 45·8-s + 35·10-s − 26·11-s − 65·14-s + 89·16-s − 77·17-s + 126·19-s − 119·20-s + 130·22-s + 96·23-s − 76·25-s + 221·28-s + 82·29-s − 196·31-s − 85·32-s + 385·34-s − 91·35-s + 131·37-s − 630·38-s + 315·40-s + 336·41-s − 201·43-s − 442·44-s + ⋯
L(s)  = 1  − 1.76·2-s + 17/8·4-s − 0.626·5-s + 0.701·7-s − 1.98·8-s + 1.10·10-s − 0.712·11-s − 1.24·14-s + 1.39·16-s − 1.09·17-s + 1.52·19-s − 1.33·20-s + 1.25·22-s + 0.870·23-s − 0.607·25-s + 1.49·28-s + 0.525·29-s − 1.13·31-s − 0.469·32-s + 1.94·34-s − 0.439·35-s + 0.582·37-s − 2.68·38-s + 1.24·40-s + 1.27·41-s − 0.712·43-s − 1.51·44-s + ⋯

Functional equation

Λ(s)=(1521s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1521s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15211521    =    32⋅1323^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 89.741989.7419
Root analytic conductor: 9.473229.47322
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1521, ( :3/2), −1)(2,\ 1521,\ (\ :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)
bad3 1 1
13 1 1
good2 1+5T+p3T2 1 + 5 T + p^{3} T^{2}
5 1+7T+p3T2 1 + 7 T + p^{3} T^{2}
7 1−13T+p3T2 1 - 13 T + p^{3} T^{2}
11 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
17 1+77T+p3T2 1 + 77 T + p^{3} T^{2}
19 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
23 1−96T+p3T2 1 - 96 T + p^{3} T^{2}
29 1−82T+p3T2 1 - 82 T + p^{3} T^{2}
31 1+196T+p3T2 1 + 196 T + p^{3} T^{2}
37 1−131T+p3T2 1 - 131 T + p^{3} T^{2}
41 1−336T+p3T2 1 - 336 T + p^{3} T^{2}
43 1+201T+p3T2 1 + 201 T + p^{3} T^{2}
47 1+105T+p3T2 1 + 105 T + p^{3} T^{2}
53 1−432T+p3T2 1 - 432 T + p^{3} T^{2}
59 1+294T+p3T2 1 + 294 T + p^{3} T^{2}
61 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
67 1+478T+p3T2 1 + 478 T + p^{3} T^{2}
71 1−9T+p3T2 1 - 9 T + p^{3} T^{2}
73 1+98T+p3T2 1 + 98 T + p^{3} T^{2}
79 1−1304T+p3T2 1 - 1304 T + p^{3} T^{2}
83 1+308T+p3T2 1 + 308 T + p^{3} T^{2}
89 1+1190T+p3T2 1 + 1190 T + p^{3} T^{2}
97 1+70T+p3T2 1 + 70 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

−8.734760012427977889802254294901, −7.87180965623879135931531860106, −7.54326851861019013554170146958, −6.73478700623421280187707382153, −5.55376550235472465605066509424, −4.51570671946648497896695462422, −3.12934645262366110467652209239, −2.10925236850159177021432798024, −1.03745381495586958894139564729, 0, 1.03745381495586958894139564729, 2.10925236850159177021432798024, 3.12934645262366110467652209239, 4.51570671946648497896695462422, 5.55376550235472465605066509424, 6.73478700623421280187707382153, 7.54326851861019013554170146958, 7.87180965623879135931531860106, 8.734760012427977889802254294901

Graph of the ZZ-function along the critical line