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Properties

Label 4-240e2-1.1-c1e2-0-9
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 yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 9-s − 25-s + 12·29-s + 12·41-s − 2·45-s + 14·49-s + 4·61-s + 81-s − 12·89-s − 36·101-s + 4·109-s − 6·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 0.894·5-s − 1/3·9-s − 1/5·25-s + 2.22·29-s + 1.87·41-s − 0.298·45-s + 2·49-s + 0.512·61-s + 1/9·81-s − 1.27·89-s − 3.58·101-s + 0.383·109-s − 0.545·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-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: yes
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 1.6257766101.625776610
L(12)L(\frac12) ≈\approx 1.6257766101.625776610
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 1−2T+pT2 1 - 2 T + p T^{2}
good7C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C22C_2^2 1−10T2+p2T4 1 - 10 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

−10.01310549089899510498567713509, −9.516767031653194228293505000223, −8.999222769477543579645116967896, −8.532532851007177915373363419493, −8.009712187101956172091092596951, −7.39696023102605292005128379490, −6.75626336855671302421092609277, −6.27483845369351885835071661051, −5.70601464750226217237219593757, −5.28260549942834099623543513745, −4.45715443970348700585757553785, −3.90964196321524344299090893356, −2.79608921257616715227246506752, −2.40914995981068008590123379595, −1.16055652003619941740790198107, 1.16055652003619941740790198107, 2.40914995981068008590123379595, 2.79608921257616715227246506752, 3.90964196321524344299090893356, 4.45715443970348700585757553785, 5.28260549942834099623543513745, 5.70601464750226217237219593757, 6.27483845369351885835071661051, 6.75626336855671302421092609277, 7.39696023102605292005128379490, 8.009712187101956172091092596951, 8.532532851007177915373363419493, 8.999222769477543579645116967896, 9.516767031653194228293505000223, 10.01310549089899510498567713509

Graph of the ZZ-function along the critical line