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Properties

Label 4-240e2-1.1-c1e2-0-0
Degree 44
Conductor 5760057600
Sign 11
Analytic cond. 3.672623.67262
Root an. cond. 1.384341.38434
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  − 4·5-s − 9-s − 4·11-s + 11·25-s + 16·31-s + 4·41-s + 4·45-s + 10·49-s + 16·55-s + 20·59-s + 4·61-s − 24·71-s + 81-s + 20·89-s + 4·99-s − 16·101-s − 20·109-s − 10·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·155-s + 157-s + ⋯
L(s)  = 1  − 1.78·5-s − 1/3·9-s − 1.20·11-s + 11/5·25-s + 2.87·31-s + 0.624·41-s + 0.596·45-s + 10/7·49-s + 2.15·55-s + 2.60·59-s + 0.512·61-s − 2.84·71-s + 1/9·81-s + 2.11·89-s + 0.402·99-s − 1.59·101-s − 1.91·109-s − 0.909·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=(57600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(57600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5760057600    =    28⋅32⋅522^{8} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 3.672623.67262
Root analytic conductor: 1.384341.38434
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 57600, ( :1/2,1/2), 1)(4,\ 57600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.78256957180.7825695718
L(12)L(\frac12) ≈\approx 0.78256957180.7825695718
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 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 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} )
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
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}
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{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

−11.97373279112430878810242545265, −11.97204191946259444215266705660, −11.65746295943836746628078584697, −10.85038935669694538881413320876, −10.60982430904786714834563426049, −10.13056002929891195176822211936, −9.499311722217538316972029861289, −8.637805473386390159780841439932, −8.456148144676839489131086932562, −7.957753317329978975262304578407, −7.53127668903399757597493950167, −7.00717100446239630133784520713, −6.41014651359295177422336904223, −5.61924101980997412333828064393, −5.02781841842898122069153631094, −4.36151973355833603134652685788, −3.94695391081871525948763714951, −2.98301548675506377793302167956, −2.58342719920451962723501294913, −0.72864190327182967399520039641, 0.72864190327182967399520039641, 2.58342719920451962723501294913, 2.98301548675506377793302167956, 3.94695391081871525948763714951, 4.36151973355833603134652685788, 5.02781841842898122069153631094, 5.61924101980997412333828064393, 6.41014651359295177422336904223, 7.00717100446239630133784520713, 7.53127668903399757597493950167, 7.957753317329978975262304578407, 8.456148144676839489131086932562, 8.637805473386390159780841439932, 9.499311722217538316972029861289, 10.13056002929891195176822211936, 10.60982430904786714834563426049, 10.85038935669694538881413320876, 11.65746295943836746628078584697, 11.97204191946259444215266705660, 11.97373279112430878810242545265

Graph of the ZZ-function along the critical line