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Properties

Label 2-58800-1.1-c1-0-163
Degree 22
Conductor 5880058800
Sign −1-1
Analytic cond. 469.520469.520
Root an. cond. 21.668421.6684
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 3·11-s + 13-s − 7·19-s − 5·23-s + 27-s + 6·31-s − 3·33-s − 3·37-s + 39-s + 3·41-s + 8·43-s + 47-s − 5·53-s − 7·57-s − 4·59-s + 8·61-s − 5·69-s + 6·71-s − 14·73-s + 16·79-s + 81-s + 16·83-s − 6·89-s + 6·93-s + 16·97-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.904·11-s + 0.277·13-s − 1.60·19-s − 1.04·23-s + 0.192·27-s + 1.07·31-s − 0.522·33-s − 0.493·37-s + 0.160·39-s + 0.468·41-s + 1.21·43-s + 0.145·47-s − 0.686·53-s − 0.927·57-s − 0.520·59-s + 1.02·61-s − 0.601·69-s + 0.712·71-s − 1.63·73-s + 1.80·79-s + 1/9·81-s + 1.75·83-s − 0.635·89-s + 0.622·93-s + 1.62·97-s + ⋯

Functional equation

Λ(s)=(58800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(58800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5880058800    =    24⋅3⋅52⋅722^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 469.520469.520
Root analytic conductor: 21.668421.6684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 58800, ( :1/2), −1)(2,\ 58800,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)Isogeny Class over Fp\mathbf{F}_p
bad2 1 1
3 1−T 1 - T
5 1 1
7 1 1
good11 1+3T+pT2 1 + 3 T + p T^{2} 1.11.d
13 1−T+pT2 1 - T + p T^{2} 1.13.ab
17 1+pT2 1 + p T^{2} 1.17.a
19 1+7T+pT2 1 + 7 T + p T^{2} 1.19.h
23 1+5T+pT2 1 + 5 T + p T^{2} 1.23.f
29 1+pT2 1 + p T^{2} 1.29.a
31 1−6T+pT2 1 - 6 T + p T^{2} 1.31.ag
37 1+3T+pT2 1 + 3 T + p T^{2} 1.37.d
41 1−3T+pT2 1 - 3 T + p T^{2} 1.41.ad
43 1−8T+pT2 1 - 8 T + p T^{2} 1.43.ai
47 1−T+pT2 1 - T + p T^{2} 1.47.ab
53 1+5T+pT2 1 + 5 T + p T^{2} 1.53.f
59 1+4T+pT2 1 + 4 T + p T^{2} 1.59.e
61 1−8T+pT2 1 - 8 T + p T^{2} 1.61.ai
67 1+pT2 1 + p T^{2} 1.67.a
71 1−6T+pT2 1 - 6 T + p T^{2} 1.71.ag
73 1+14T+pT2 1 + 14 T + p T^{2} 1.73.o
79 1−16T+pT2 1 - 16 T + p T^{2} 1.79.aq
83 1−16T+pT2 1 - 16 T + p T^{2} 1.83.aq
89 1+6T+pT2 1 + 6 T + p T^{2} 1.89.g
97 1−16T+pT2 1 - 16 T + p T^{2} 1.97.aq
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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

−14.57955247889153, −14.05398339041032, −13.58620690171007, −13.15388636292091, −12.52225457989181, −12.30135408401430, −11.50697065008264, −10.90122883060256, −10.43602563626131, −10.10904977501148, −9.360209011441947, −8.909059807505478, −8.234782906619250, −8.001918321340518, −7.426445939327318, −6.644860964588996, −6.216955475616820, −5.629111242055402, −4.846784487106846, −4.330941396444921, −3.782156724526464, −3.058476887834843, −2.326218990148043, −2.008419252823793, −0.9350577989650694, 0, 0.9350577989650694, 2.008419252823793, 2.326218990148043, 3.058476887834843, 3.782156724526464, 4.330941396444921, 4.846784487106846, 5.629111242055402, 6.216955475616820, 6.644860964588996, 7.426445939327318, 8.001918321340518, 8.234782906619250, 8.909059807505478, 9.360209011441947, 10.10904977501148, 10.43602563626131, 10.90122883060256, 11.50697065008264, 12.30135408401430, 12.52225457989181, 13.15388636292091, 13.58620690171007, 14.05398339041032, 14.57955247889153

Graph of the ZZ-function along the critical line