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Properties

Label 4-40e4-1.1-c1e2-0-40
Degree 44
Conductor 25600002560000
Sign 11
Analytic cond. 163.227163.227
Root an. cond. 3.574363.57436
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  − 2·3-s − 4·7-s + 2·9-s − 2·11-s + 2·13-s + 6·19-s + 8·21-s − 12·23-s − 6·27-s − 6·29-s + 16·31-s + 4·33-s − 6·37-s − 4·39-s − 10·43-s − 2·49-s − 10·53-s − 12·57-s − 6·59-s − 18·61-s − 8·63-s + 10·67-s + 24·69-s − 8·73-s + 8·77-s + 11·81-s − 2·83-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 2/3·9-s − 0.603·11-s + 0.554·13-s + 1.37·19-s + 1.74·21-s − 2.50·23-s − 1.15·27-s − 1.11·29-s + 2.87·31-s + 0.696·33-s − 0.986·37-s − 0.640·39-s − 1.52·43-s − 2/7·49-s − 1.37·53-s − 1.58·57-s − 0.781·59-s − 2.30·61-s − 1.00·63-s + 1.22·67-s + 2.88·69-s − 0.936·73-s + 0.911·77-s + 11/9·81-s − 0.219·83-s + ⋯

Functional equation

Λ(s)=(2560000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2560000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 25600002560000    =    212⋅542^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 163.227163.227
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 2560000, ( :1/2,1/2), 1)(4,\ 2560000,\ (\ :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
5 1 1
good3C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
7C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
11C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1−6T+pT2)(1+4T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 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}
23C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
29C2C_2 (1−4T+pT2)(1+10T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
41C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
43C22C_2^2 1+10T+50T2+10pT3+p2T4 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
53C2C_2 (1−4T+pT2)(1+14T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 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}
67C22C_2^2 1−10T+50T2−10pT3+p2T4 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−42T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
73C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
89C22C_2^2 1−162T2+p2T4 1 - 162 T^{2} + p^{2} T^{4}
97C22C_2^2 1−190T2+p2T4 1 - 190 T^{2} + 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

−9.409849107021290924212913667598, −8.973240765902173635808131042840, −8.166221422442874923142527804299, −8.021975407540844688834874989070, −7.69637422811741651850934558385, −7.10033155884986463762739968618, −6.47840116003058930629506805179, −6.42877478120596479587860185234, −5.87631680476208146052773780967, −5.82865531445608805589229308253, −5.00401420807203998386668496800, −4.86788396244498416072826967746, −4.09648511318576568001315043004, −3.66254060472490055928832640638, −3.18816530161154160413273479350, −2.77741016294620158090345512097, −1.87310610378703042054930399652, −1.28971217048944487230143332373, 0, 0, 1.28971217048944487230143332373, 1.87310610378703042054930399652, 2.77741016294620158090345512097, 3.18816530161154160413273479350, 3.66254060472490055928832640638, 4.09648511318576568001315043004, 4.86788396244498416072826967746, 5.00401420807203998386668496800, 5.82865531445608805589229308253, 5.87631680476208146052773780967, 6.42877478120596479587860185234, 6.47840116003058930629506805179, 7.10033155884986463762739968618, 7.69637422811741651850934558385, 8.021975407540844688834874989070, 8.166221422442874923142527804299, 8.973240765902173635808131042840, 9.409849107021290924212913667598

Graph of the ZZ-function along the critical line