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Properties

Label 4-240e2-1.1-c1e2-0-8
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  + 9-s + 8·23-s − 25-s + 8·31-s − 4·41-s + 8·47-s − 10·49-s + 16·71-s − 4·73-s + 8·79-s + 81-s + 12·89-s − 4·97-s + 8·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 1/3·9-s + 1.66·23-s − 1/5·25-s + 1.43·31-s − 0.624·41-s + 1.16·47-s − 1.42·49-s + 1.89·71-s − 0.468·73-s + 0.900·79-s + 1/9·81-s + 1.27·89-s − 0.406·97-s + 0.752·113-s − 0.181·121-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 + 6/13·169-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 1.4614501321.461450132
L(12)L(\frac12) ≈\approx 1.4614501321.461450132
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×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5C2C_2 1+T2 1 + T^{2}
good7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
13C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
23C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
29C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
31C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
37C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
61C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−16T+pT2)(1+pT2) ( 1 - 16 T + p T^{2} )( 1 + p T^{2} )
73C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
79C2C_2×\timesC2C_2 (1−16T+pT2)(1+8T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−14T+pT2)(1+2T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−14T+pT2)(1+18T+pT2) ( 1 - 14 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

−10.06325448658857273904204313022, −9.402417311024011038569025377487, −9.076947808228994683297428556955, −8.448702233738562370315415025932, −7.942716130182528340399222636327, −7.44867711041723846797068290753, −6.70244626183064586674998156022, −6.52769857784694909597587762189, −5.68344208303553593197376005547, −5.00465741226719856557315685797, −4.62258139326400856372813398736, −3.76161657111378754289371065947, −3.09464338182030477917379853018, −2.27117761095347414625560142043, −1.09137919815179101298622070951, 1.09137919815179101298622070951, 2.27117761095347414625560142043, 3.09464338182030477917379853018, 3.76161657111378754289371065947, 4.62258139326400856372813398736, 5.00465741226719856557315685797, 5.68344208303553593197376005547, 6.52769857784694909597587762189, 6.70244626183064586674998156022, 7.44867711041723846797068290753, 7.942716130182528340399222636327, 8.448702233738562370315415025932, 9.076947808228994683297428556955, 9.402417311024011038569025377487, 10.06325448658857273904204313022

Graph of the ZZ-function along the critical line