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Properties

Label 4-5712e2-1.1-c1e2-0-3
Degree 44
Conductor 3262694432626944
Sign 11
Analytic cond. 2080.322080.32
Root an. cond. 6.753556.75355
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  − 2·3-s + 4·5-s − 2·7-s + 3·9-s − 4·13-s − 8·15-s − 2·17-s + 4·21-s + 8·23-s + 2·25-s − 4·27-s + 4·29-s − 8·35-s − 4·37-s + 8·39-s − 4·41-s − 8·43-s + 12·45-s + 16·47-s + 3·49-s + 4·51-s + 12·53-s + 8·59-s + 20·61-s − 6·63-s − 16·65-s + 8·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s − 0.755·7-s + 9-s − 1.10·13-s − 2.06·15-s − 0.485·17-s + 0.872·21-s + 1.66·23-s + 2/5·25-s − 0.769·27-s + 0.742·29-s − 1.35·35-s − 0.657·37-s + 1.28·39-s − 0.624·41-s − 1.21·43-s + 1.78·45-s + 2.33·47-s + 3/7·49-s + 0.560·51-s + 1.64·53-s + 1.04·59-s + 2.56·61-s − 0.755·63-s − 1.98·65-s + 0.977·67-s + ⋯

Functional equation

Λ(s)=(32626944s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 32626944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(32626944s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 32626944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3262694432626944    =    28⋅32⋅72⋅1722^{8} \cdot 3^{2} \cdot 7^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 2080.322080.32
Root analytic conductor: 6.753556.75355
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 32626944, ( :1/2,1/2), 1)(4,\ 32626944,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4123782172.412378217
L(12)L(\frac12) ≈\approx 2.4123782172.412378217
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
3C1C_1 (1+T)2 ( 1 + T )^{2}
7C1C_1 (1+T)2 ( 1 + T )^{2}
17C1C_1 (1+T)2 ( 1 + T )^{2}
good5C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
13D4D_{4} 1+4T+6T2+4pT3+p2T4 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
23C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
29D4D_{4} 1−4T+38T2−4pT3+p2T4 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4}
31C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
37D4D_{4} 1+4T+54T2+4pT3+p2T4 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59D4D_{4} 1−8T+110T2−8pT3+p2T4 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4}
61C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
67D4D_{4} 1−8T+54T2−8pT3+p2T4 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+8T+62T2+8pT3+p2T4 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4}
73C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
79C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
83D4D_{4} 1−24T+286T2−24pT3+p2T4 1 - 24 T + 286 T^{2} - 24 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+4T+86T2+4pT3+p2T4 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4}
97C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
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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

−8.401623292797585664846043336208, −7.88410839683485929838060750231, −7.30355865638477833800684086370, −7.03678399563654217145024819803, −6.83634885094482850788032048503, −6.51738791382209598532333492140, −6.20226460058585050195656426300, −5.65248135210590006836623387310, −5.37695213827609845087914980861, −5.36781726853962615356005390727, −4.84468347416322590841025997214, −4.45085121858457059840735139381, −3.81614883294576252887817220802, −3.61180129386281851974508791446, −2.74040109881289772290551498915, −2.57326782856774654495500389038, −2.06218888505628783581297165438, −1.69942441547186803777852439627, −0.864076720247538904497898252984, −0.54188328923494244407794846359, 0.54188328923494244407794846359, 0.864076720247538904497898252984, 1.69942441547186803777852439627, 2.06218888505628783581297165438, 2.57326782856774654495500389038, 2.74040109881289772290551498915, 3.61180129386281851974508791446, 3.81614883294576252887817220802, 4.45085121858457059840735139381, 4.84468347416322590841025997214, 5.36781726853962615356005390727, 5.37695213827609845087914980861, 5.65248135210590006836623387310, 6.20226460058585050195656426300, 6.51738791382209598532333492140, 6.83634885094482850788032048503, 7.03678399563654217145024819803, 7.30355865638477833800684086370, 7.88410839683485929838060750231, 8.401623292797585664846043336208

Graph of the ZZ-function along the critical line