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Properties

Label 4-3960e2-1.1-c1e2-0-14
Degree 44
Conductor 1568160015681600
Sign 11
Analytic cond. 999.872999.872
Root an. cond. 5.623235.62323
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·11-s − 4·13-s − 4·17-s + 3·25-s − 4·29-s − 4·31-s − 4·37-s − 4·41-s + 8·43-s − 12·49-s − 20·53-s − 4·55-s − 4·59-s − 12·61-s − 8·65-s − 12·71-s + 12·73-s − 8·79-s − 8·83-s − 8·85-s − 20·89-s − 4·97-s + 4·101-s + 24·103-s − 8·107-s − 4·109-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.603·11-s − 1.10·13-s − 0.970·17-s + 3/5·25-s − 0.742·29-s − 0.718·31-s − 0.657·37-s − 0.624·41-s + 1.21·43-s − 1.71·49-s − 2.74·53-s − 0.539·55-s − 0.520·59-s − 1.53·61-s − 0.992·65-s − 1.42·71-s + 1.40·73-s − 0.900·79-s − 0.878·83-s − 0.867·85-s − 2.11·89-s − 0.406·97-s + 0.398·101-s + 2.36·103-s − 0.773·107-s − 0.383·109-s + ⋯

Functional equation

Λ(s)=(15681600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(15681600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1568160015681600    =    26⋅34⋅52⋅1122^{6} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 999.872999.872
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 15681600, ( :1/2,1/2), 1)(4,\ 15681600,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
11C1C_1 (1+T)2 ( 1 + T )^{2}
good7C22C_2^2 1+12T2+p2T4 1 + 12 T^{2} + p^{2} T^{4}
13D4D_{4} 1+4T+28T2+4pT3+p2T4 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+4T+36T2+4pT3+p2T4 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
23C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
29D4D_{4} 1+4T+54T2+4pT3+p2T4 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}
31C4C_4 1+4T−6T2+4pT3+p2T4 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+4T+46T2+4pT3+p2T4 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+4T+14T2+4pT3+p2T4 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−8T+100T2−8pT3+p2T4 1 - 8 T + 100 T^{2} - 8 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+86T2+p2T4 1 + 86 T^{2} + p^{2} T^{4}
53D4D_{4} 1+20T+198T2+20pT3+p2T4 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+4T+50T2+4pT3+p2T4 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+12T+150T2+12pT3+p2T4 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
71C4C_4 1+12T+170T2+12pT3+p2T4 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−12T+180T2−12pT3+p2T4 1 - 12 T + 180 T^{2} - 12 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+8T+142T2+8pT3+p2T4 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+8T+164T2+8pT3+p2T4 1 + 8 T + 164 T^{2} + 8 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+20T+246T2+20pT3+p2T4 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+4T+190T2+4pT3+p2T4 1 + 4 T + 190 T^{2} + 4 p T^{3} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.265513164557404834395079030213, −7.908970079033027260109667651765, −7.37982476357739626880923717120, −7.36232082944698600963901364124, −6.70625806993815846572572581033, −6.51074593252933263327192280547, −6.00566937903738438986698599382, −5.70372082677681664040108252280, −5.28057167260991902992444616535, −4.89672981378701668148221215481, −4.52255745184981330663156890322, −4.30217181647171831732431033724, −3.36950202238629285004491334713, −3.26759731766603956092872859718, −2.57361798734579017977230528527, −2.32677666498597041577469461569, −1.61609576815564888925979769689, −1.46443394382296823242613516156, 0, 0, 1.46443394382296823242613516156, 1.61609576815564888925979769689, 2.32677666498597041577469461569, 2.57361798734579017977230528527, 3.26759731766603956092872859718, 3.36950202238629285004491334713, 4.30217181647171831732431033724, 4.52255745184981330663156890322, 4.89672981378701668148221215481, 5.28057167260991902992444616535, 5.70372082677681664040108252280, 6.00566937903738438986698599382, 6.51074593252933263327192280547, 6.70625806993815846572572581033, 7.36232082944698600963901364124, 7.37982476357739626880923717120, 7.908970079033027260109667651765, 8.265513164557404834395079030213

Graph of the ZZ-function along the critical line