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Properties

Label 2-4304-1.1-c1-0-125
Degree 22
Conductor 43044304
Sign −1-1
Analytic cond. 34.367634.3676
Root an. cond. 5.862385.86238
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.48·3-s + 3.18·5-s − 3.31·7-s − 0.793·9-s − 0.867·11-s − 2.69·13-s + 4.73·15-s + 2.03·17-s − 8.43·19-s − 4.91·21-s + 6.66·23-s + 5.14·25-s − 5.63·27-s − 7.02·29-s − 4.72·31-s − 1.28·33-s − 10.5·35-s − 1.00·37-s − 4.00·39-s + 11.7·41-s − 1.30·43-s − 2.52·45-s − 5.57·47-s + 3.97·49-s + 3.02·51-s − 5.95·53-s − 2.76·55-s + ⋯
L(s)  = 1  + 0.857·3-s + 1.42·5-s − 1.25·7-s − 0.264·9-s − 0.261·11-s − 0.747·13-s + 1.22·15-s + 0.493·17-s − 1.93·19-s − 1.07·21-s + 1.38·23-s + 1.02·25-s − 1.08·27-s − 1.30·29-s − 0.848·31-s − 0.224·33-s − 1.78·35-s − 0.165·37-s − 0.640·39-s + 1.82·41-s − 0.199·43-s − 0.376·45-s − 0.813·47-s + 0.567·49-s + 0.423·51-s − 0.817·53-s − 0.372·55-s + ⋯

Functional equation

Λ(s)=(4304s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4304s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43044304    =    24⋅2692^{4} \cdot 269
Sign: −1-1
Analytic conductor: 34.367634.3676
Root analytic conductor: 5.862385.86238
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4304, ( :1/2), −1)(2,\ 4304,\ (\ :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
269 1−T 1 - T
good3 1−1.48T+3T2 1 - 1.48T + 3T^{2}
5 1−3.18T+5T2 1 - 3.18T + 5T^{2}
7 1+3.31T+7T2 1 + 3.31T + 7T^{2}
11 1+0.867T+11T2 1 + 0.867T + 11T^{2}
13 1+2.69T+13T2 1 + 2.69T + 13T^{2}
17 1−2.03T+17T2 1 - 2.03T + 17T^{2}
19 1+8.43T+19T2 1 + 8.43T + 19T^{2}
23 1−6.66T+23T2 1 - 6.66T + 23T^{2}
29 1+7.02T+29T2 1 + 7.02T + 29T^{2}
31 1+4.72T+31T2 1 + 4.72T + 31T^{2}
37 1+1.00T+37T2 1 + 1.00T + 37T^{2}
41 1−11.7T+41T2 1 - 11.7T + 41T^{2}
43 1+1.30T+43T2 1 + 1.30T + 43T^{2}
47 1+5.57T+47T2 1 + 5.57T + 47T^{2}
53 1+5.95T+53T2 1 + 5.95T + 53T^{2}
59 1+9.60T+59T2 1 + 9.60T + 59T^{2}
61 1−5.70T+61T2 1 - 5.70T + 61T^{2}
67 1−4.86T+67T2 1 - 4.86T + 67T^{2}
71 1−2.44T+71T2 1 - 2.44T + 71T^{2}
73 1−0.698T+73T2 1 - 0.698T + 73T^{2}
79 1−3.24T+79T2 1 - 3.24T + 79T^{2}
83 1+15.3T+83T2 1 + 15.3T + 83T^{2}
89 1+6.06T+89T2 1 + 6.06T + 89T^{2}
97 1+7.76T+97T2 1 + 7.76T + 97T^{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

−8.132990615005969296661931998879, −7.23112832077786520343468118403, −6.51611127601154433720487330687, −5.85351250745129015785207560926, −5.23374110035147950287181168966, −4.05867125238433398732176349523, −3.07952971693879330318221056637, −2.54168558773085624465513847514, −1.76941416862055347408073050612, 0, 1.76941416862055347408073050612, 2.54168558773085624465513847514, 3.07952971693879330318221056637, 4.05867125238433398732176349523, 5.23374110035147950287181168966, 5.85351250745129015785207560926, 6.51611127601154433720487330687, 7.23112832077786520343468118403, 8.132990615005969296661931998879

Graph of the ZZ-function along the critical line