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Properties

Label 2-39e2-1.1-c3-0-128
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  − 2-s − 7·4-s + 7·5-s + 10·7-s + 15·8-s − 7·10-s − 22·11-s − 10·14-s + 41·16-s − 37·17-s − 30·19-s − 49·20-s + 22·22-s + 162·23-s − 76·25-s − 70·28-s + 113·29-s − 196·31-s − 161·32-s + 37·34-s + 70·35-s − 13·37-s + 30·38-s + 105·40-s + 285·41-s − 246·43-s + 154·44-s + ⋯
L(s)  = 1  − 0.353·2-s − 7/8·4-s + 0.626·5-s + 0.539·7-s + 0.662·8-s − 0.221·10-s − 0.603·11-s − 0.190·14-s + 0.640·16-s − 0.527·17-s − 0.362·19-s − 0.547·20-s + 0.213·22-s + 1.46·23-s − 0.607·25-s − 0.472·28-s + 0.723·29-s − 1.13·31-s − 0.889·32-s + 0.186·34-s + 0.338·35-s − 0.0577·37-s + 0.128·38-s + 0.415·40-s + 1.08·41-s − 0.872·43-s + 0.527·44-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+T+p3T2 1 + T + p^{3} T^{2}
5 1−7T+p3T2 1 - 7 T + p^{3} T^{2}
7 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
11 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
17 1+37T+p3T2 1 + 37 T + p^{3} T^{2}
19 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
23 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
29 1−113T+p3T2 1 - 113 T + p^{3} T^{2}
31 1+196T+p3T2 1 + 196 T + p^{3} T^{2}
37 1+13T+p3T2 1 + 13 T + p^{3} T^{2}
41 1−285T+p3T2 1 - 285 T + p^{3} T^{2}
43 1+246T+p3T2 1 + 246 T + p^{3} T^{2}
47 1+462T+p3T2 1 + 462 T + p^{3} T^{2}
53 1−537T+p3T2 1 - 537 T + p^{3} T^{2}
59 1−576T+p3T2 1 - 576 T + p^{3} T^{2}
61 1+635T+p3T2 1 + 635 T + p^{3} T^{2}
67 1+202T+p3T2 1 + 202 T + p^{3} T^{2}
71 1+1086T+p3T2 1 + 1086 T + p^{3} T^{2}
73 1−805T+p3T2 1 - 805 T + p^{3} T^{2}
79 1−884T+p3T2 1 - 884 T + p^{3} T^{2}
83 1−518T+p3T2 1 - 518 T + p^{3} T^{2}
89 1−194T+p3T2 1 - 194 T + p^{3} T^{2}
97 1−1202T+p3T2 1 - 1202 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

−8.803756649813499893023455886656, −8.053417034673223902307939725107, −7.26599330844971751653116856537, −6.19448945249893137522473985733, −5.18865668346464858569492939138, −4.72256343699449993591989685845, −3.57650642168202236891237943258, −2.30233839815696874325824130612, −1.24403047013894068968848515494, 0, 1.24403047013894068968848515494, 2.30233839815696874325824130612, 3.57650642168202236891237943258, 4.72256343699449993591989685845, 5.18865668346464858569492939138, 6.19448945249893137522473985733, 7.26599330844971751653116856537, 8.053417034673223902307939725107, 8.803756649813499893023455886656

Graph of the ZZ-function along the critical line