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Properties

Label 4-240e2-1.1-c1e2-0-42
Degree 44
Conductor 5760057600
Sign −1-1
Analytic cond. 3.672623.67262
Root an. cond. 1.384341.38434
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 4·11-s − 8·19-s − 5·25-s − 8·29-s − 8·31-s + 4·41-s − 6·49-s + 4·59-s + 4·61-s + 24·71-s + 8·79-s + 81-s + 4·89-s + 4·99-s − 8·101-s − 4·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.20·11-s − 1.83·19-s − 25-s − 1.48·29-s − 1.43·31-s + 0.624·41-s − 6/7·49-s + 0.520·59-s + 0.512·61-s + 2.84·71-s + 0.900·79-s + 1/9·81-s + 0.423·89-s + 0.402·99-s − 0.796·101-s − 0.383·109-s + 6/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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: −1-1
Analytic conductor: 3.672623.67262
Root analytic conductor: 1.384341.38434
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 57600, ( :1/2,1/2), −1)(4,\ 57600,\ (\ :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
3C2C_2 1+T2 1 + T^{2}
5C2C_2 1+pT2 1 + p T^{2}
good7C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
11C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
23C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
29C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
31C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
37C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
47C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
53C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−6T+pT2)(1+2T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{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+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
83C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C22C_2^2 1+62T2+p2T4 1 + 62 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

−9.611140669062864815271056558566, −9.365889598328798516671123406442, −8.615144406860878781627449204763, −8.214329307238486447538791212556, −7.72000825758485814981796044212, −7.25156720215625820833061588504, −6.49966207063261834816622982659, −6.01230749466635234945297225244, −5.37059153541267463388606524787, −4.97032745406773201180690943389, −3.98573809930382561611866429864, −3.61747823503815992595013485966, −2.46656285877604068153338452679, −1.99173296633949866898224452290, 0, 1.99173296633949866898224452290, 2.46656285877604068153338452679, 3.61747823503815992595013485966, 3.98573809930382561611866429864, 4.97032745406773201180690943389, 5.37059153541267463388606524787, 6.01230749466635234945297225244, 6.49966207063261834816622982659, 7.25156720215625820833061588504, 7.72000825758485814981796044212, 8.214329307238486447538791212556, 8.615144406860878781627449204763, 9.365889598328798516671123406442, 9.611140669062864815271056558566

Graph of the ZZ-function along the critical line