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Properties

Label 4-1920e2-1.1-c1e2-0-1
Degree 44
Conductor 36864003686400
Sign 11
Analytic cond. 235.048235.048
Root an. cond. 3.915513.91551
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 6·7-s − 9-s + 2·11-s − 8·13-s + 6·17-s − 6·19-s + 2·23-s + 11·25-s − 6·29-s − 24·35-s − 16·37-s − 12·43-s − 4·45-s − 10·47-s + 18·49-s + 8·55-s − 14·59-s + 18·61-s + 6·63-s − 32·65-s − 4·67-s + 16·71-s − 10·73-s − 12·77-s + 81-s + 24·85-s + ⋯
L(s)  = 1  + 1.78·5-s − 2.26·7-s − 1/3·9-s + 0.603·11-s − 2.21·13-s + 1.45·17-s − 1.37·19-s + 0.417·23-s + 11/5·25-s − 1.11·29-s − 4.05·35-s − 2.63·37-s − 1.82·43-s − 0.596·45-s − 1.45·47-s + 18/7·49-s + 1.07·55-s − 1.82·59-s + 2.30·61-s + 0.755·63-s − 3.96·65-s − 0.488·67-s + 1.89·71-s − 1.17·73-s − 1.36·77-s + 1/9·81-s + 2.60·85-s + ⋯

Functional equation

Λ(s)=(3686400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3686400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 36864003686400    =    214⋅32⋅522^{14} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 235.048235.048
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 3686400, ( :1/2,1/2), 1)(4,\ 3686400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.62429290820.6242929082
L(12)L(\frac12) ≈\approx 0.62429290820.6242929082
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
3C2C_2 1+T2 1 + T^{2}
5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good7C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
17C2C_2 (1−8T+pT2)(1+2T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
29C2C_2 (1−4T+pT2)(1+10T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
43C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
47C22C_2^2 1+10T+50T2+10pT3+p2T4 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C22C_2^2 1+14T+98T2+14pT3+p2T4 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−18T+162T2−18pT3+p2T4 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4}
67C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+16T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
97C22C_2^2 1+14T+98T2+14pT3+p2T4 1 + 14 T + 98 T^{2} + 14 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

−9.653976858942879259257762436987, −9.106309189021959827167912669434, −8.917837311603604779674577775645, −8.267205873821144939363906250142, −7.891379922770582023928293534658, −7.04856826438382263232025938415, −6.98010916421581563782250907528, −6.56191891001832729600629776126, −6.43181830637520079300615743134, −5.72088047262283426678519189245, −5.50326203304008999628408397645, −5.07952247830461095011450602086, −4.73433350620126477554795720053, −3.74902053988551720427260817106, −3.50844155819926030731595601368, −2.93736733401200392927571435665, −2.63394281614110867228576864515, −1.94714286290854385696877503072, −1.56146344371270900377637906142, −0.26541719914254131492948166175, 0.26541719914254131492948166175, 1.56146344371270900377637906142, 1.94714286290854385696877503072, 2.63394281614110867228576864515, 2.93736733401200392927571435665, 3.50844155819926030731595601368, 3.74902053988551720427260817106, 4.73433350620126477554795720053, 5.07952247830461095011450602086, 5.50326203304008999628408397645, 5.72088047262283426678519189245, 6.43181830637520079300615743134, 6.56191891001832729600629776126, 6.98010916421581563782250907528, 7.04856826438382263232025938415, 7.891379922770582023928293534658, 8.267205873821144939363906250142, 8.917837311603604779674577775645, 9.106309189021959827167912669434, 9.653976858942879259257762436987

Graph of the ZZ-function along the critical line