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Properties

Label 4-80e3-1.1-c1e2-0-8
Degree 44
Conductor 512000512000
Sign 11
Analytic cond. 32.645532.6455
Root an. cond. 2.390312.39031
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  + 5-s + 6·9-s + 25-s + 4·29-s + 12·41-s + 6·45-s − 2·49-s − 4·61-s + 27·81-s − 12·89-s − 12·101-s + 28·109-s − 6·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 0.447·5-s + 2·9-s + 1/5·25-s + 0.742·29-s + 1.87·41-s + 0.894·45-s − 2/7·49-s − 0.512·61-s + 3·81-s − 1.27·89-s − 1.19·101-s + 2.68·109-s − 0.545·121-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.332·145-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)=(512000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 512000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(512000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 512000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 512000512000    =    212⋅532^{12} \cdot 5^{3}
Sign: 11
Analytic conductor: 32.645532.6455
Root analytic conductor: 2.390312.39031
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 512000, ( :1/2,1/2), 1)(4,\ 512000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6804989932.680498993
L(12)L(\frac12) ≈\approx 2.6804989932.680498993
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
5C1C_1 1−T 1 - T
good3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
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 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
29C2C_2 (1−2T+pT2)2 ( 1 - 2 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 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
47C22C_2^2 1−78T2+p2T4 1 - 78 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 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 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1+90T2+p2T4 1 + 90 T^{2} + p^{2} T^{4}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 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

−8.596990511932736222144817015338, −7.907341477090996490882333551725, −7.52503145482899774991079550906, −7.22991414319633251811158852213, −6.62065588307934500428137551484, −6.34353768083929153204953611946, −5.76584931651134752447239181230, −5.15483711704418209300479511955, −4.62444226502693542608431003768, −4.25628687112004479506978652093, −3.76293337439488997521798166674, −2.98579170941525266723249785242, −2.30624557105481912795372074779, −1.60242124233434574166321397201, −0.950642266431141237476176657803, 0.950642266431141237476176657803, 1.60242124233434574166321397201, 2.30624557105481912795372074779, 2.98579170941525266723249785242, 3.76293337439488997521798166674, 4.25628687112004479506978652093, 4.62444226502693542608431003768, 5.15483711704418209300479511955, 5.76584931651134752447239181230, 6.34353768083929153204953611946, 6.62065588307934500428137551484, 7.22991414319633251811158852213, 7.52503145482899774991079550906, 7.907341477090996490882333551725, 8.596990511932736222144817015338

Graph of the ZZ-function along the critical line