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Properties

Label 4-3960e2-1.1-c1e2-0-9
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 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 5·7-s + 2·11-s + 6·13-s + 3·17-s + 7·19-s + 6·23-s + 3·25-s − 13·29-s − 7·31-s + 10·35-s + 19·37-s + 8·41-s + 2·43-s − 2·47-s + 9·49-s − 7·53-s + 4·55-s + 6·59-s − 21·61-s + 12·65-s + 5·71-s − 6·73-s + 10·77-s + 2·79-s + 12·83-s + 6·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.88·7-s + 0.603·11-s + 1.66·13-s + 0.727·17-s + 1.60·19-s + 1.25·23-s + 3/5·25-s − 2.41·29-s − 1.25·31-s + 1.69·35-s + 3.12·37-s + 1.24·41-s + 0.304·43-s − 0.291·47-s + 9/7·49-s − 0.961·53-s + 0.539·55-s + 0.781·59-s − 2.68·61-s + 1.48·65-s + 0.593·71-s − 0.702·73-s + 1.13·77-s + 0.225·79-s + 1.31·83-s + 0.650·85-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: 00
Selberg data: (4, 15681600, ( :1/2,1/2), 1)(4,\ 15681600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 7.2569996147.256999614
L(12)L(\frac12) ≈\approx 7.2569996147.256999614
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}
good7D4D_{4} 1−5T+16T2−5pT3+p2T4 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4}
13C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−3T−2T2−3pT3+p2T4 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4}
19C4C_4 1−7T+46T2−7pT3+p2T4 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−6T+38T2−6pT3+p2T4 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+13T+96T2+13pT3+p2T4 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+7T+70T2+7pT3+p2T4 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−19T+160T2−19pT3+p2T4 1 - 19 T + 160 T^{2} - 19 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−8T+30T2−8pT3+p2T4 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−2T+70T2−2pT3+p2T4 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+2T−58T2+2pT3+p2T4 1 + 2 T - 58 T^{2} + 2 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+7T+80T2+7pT3+p2T4 1 + 7 T + 80 T^{2} + 7 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−6T+110T2−6pT3+p2T4 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+21T+228T2+21pT3+p2T4 1 + 21 T + 228 T^{2} + 21 p T^{3} + p^{2} T^{4}
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71D4D_{4} 1−5T+110T2−5pT3+p2T4 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+6T+138T2+6pT3+p2T4 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−2T+6T2−2pT3+p2T4 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4}
83C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
89D4D_{4} 1+7T+152T2+7pT3+p2T4 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+18T+258T2+18pT3+p2T4 1 + 18 T + 258 T^{2} + 18 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

−8.549764596071347884164245178450, −8.317083417966088551842787040030, −7.77518884452663127012928617492, −7.62552759773609875676005825338, −7.37225970981943262944471597515, −6.86602568759394421452680807508, −6.27777608560631956516505333040, −5.90141708557723876941097491007, −5.60932882720283261153447052646, −5.53876293198186659190058568918, −4.73962338589617553200782360279, −4.72311044467438401711539546030, −3.98934958251338811463628113308, −3.76093671397636364883489823639, −3.06504237899985161399957629674, −2.86281881307806943522262508550, −1.82931664528933569105866262624, −1.81394371783562424830942676410, −1.09133119145956640782465470314, −0.983009697956838234627739818625, 0.983009697956838234627739818625, 1.09133119145956640782465470314, 1.81394371783562424830942676410, 1.82931664528933569105866262624, 2.86281881307806943522262508550, 3.06504237899985161399957629674, 3.76093671397636364883489823639, 3.98934958251338811463628113308, 4.72311044467438401711539546030, 4.73962338589617553200782360279, 5.53876293198186659190058568918, 5.60932882720283261153447052646, 5.90141708557723876941097491007, 6.27777608560631956516505333040, 6.86602568759394421452680807508, 7.37225970981943262944471597515, 7.62552759773609875676005825338, 7.77518884452663127012928617492, 8.317083417966088551842787040030, 8.549764596071347884164245178450

Graph of the ZZ-function along the critical line