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Properties

Label 2-420e2-1.1-c1-0-178
Degree 22
Conductor 176400176400
Sign −1-1
Analytic cond. 1408.561408.56
Root an. cond. 37.530837.5308
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  − 4·11-s − 6·13-s + 2·17-s − 8·19-s − 4·23-s − 6·29-s + 4·31-s + 2·37-s − 2·41-s − 12·43-s + 2·53-s + 4·59-s − 6·61-s − 4·67-s + 8·71-s + 6·73-s + 16·79-s − 4·83-s − 18·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s − 1.66·13-s + 0.485·17-s − 1.83·19-s − 0.834·23-s − 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.312·41-s − 1.82·43-s + 0.274·53-s + 0.520·59-s − 0.768·61-s − 0.488·67-s + 0.949·71-s + 0.702·73-s + 1.80·79-s − 0.439·83-s − 1.90·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

Λ(s)=(176400s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(176400s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 176400176400    =    24⋅32⋅52⋅722^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 1408.561408.56
Root analytic conductor: 37.530837.5308
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 176400, ( :1/2), −1)(2,\ 176400,\ (\ :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)
bad2 1 1
3 1 1
5 1 1
7 1 1
good11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+18T+pT2 1 + 18 T + p T^{2}
97 1−6T+pT2 1 - 6 T + p 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

−13.29989630081279, −12.88817925524109, −12.51792923310917, −12.13521241601464, −11.48801870895328, −11.12330411536522, −10.38363176547975, −10.06287920873662, −9.907524589797490, −9.111260478667703, −8.574022328210184, −8.079100984757359, −7.664832714625015, −7.275312997746144, −6.552714641811738, −6.181406340075739, −5.486648871962378, −4.982903186648741, −4.661615197743061, −3.971748798082462, −3.351122316033558, −2.668425604762867, −2.150683542552333, −1.809817461518069, −0.5521525087696206, 0, 0.5521525087696206, 1.809817461518069, 2.150683542552333, 2.668425604762867, 3.351122316033558, 3.971748798082462, 4.661615197743061, 4.982903186648741, 5.486648871962378, 6.181406340075739, 6.552714641811738, 7.275312997746144, 7.664832714625015, 8.079100984757359, 8.574022328210184, 9.111260478667703, 9.907524589797490, 10.06287920873662, 10.38363176547975, 11.12330411536522, 11.48801870895328, 12.13521241601464, 12.51792923310917, 12.88817925524109, 13.29989630081279

Graph of the ZZ-function along the critical line