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Properties

Label 4-160e2-1.1-c1e2-0-8
Degree 44
Conductor 2560025600
Sign 11
Analytic cond. 1.632271.63227
Root an. cond. 1.130311.13031
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 + 2·9-s + 4·11-s − 4·19-s − 25-s + 4·29-s − 16·31-s + 4·41-s + 4·45-s − 6·49-s + 8·55-s + 20·59-s − 12·61-s + 8·71-s − 5·81-s + 4·89-s − 8·95-s + 8·99-s − 12·101-s + 20·109-s − 6·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯
L(s)  = 1  + 0.894·5-s + 2/3·9-s + 1.20·11-s − 0.917·19-s − 1/5·25-s + 0.742·29-s − 2.87·31-s + 0.624·41-s + 0.596·45-s − 6/7·49-s + 1.07·55-s + 2.60·59-s − 1.53·61-s + 0.949·71-s − 5/9·81-s + 0.423·89-s − 0.820·95-s + 0.804·99-s − 1.19·101-s + 1.91·109-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 + 0.664·145-s + ⋯

Functional equation

Λ(s)=(25600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(25600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2560025600    =    210⋅522^{10} \cdot 5^{2}
Sign: 11
Analytic conductor: 1.632271.63227
Root analytic conductor: 1.130311.13031
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 25600, ( :1/2,1/2), 1)(4,\ 25600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4945074971.494507497
L(12)L(\frac12) ≈\approx 1.4945074971.494507497
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
5C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
11C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
13C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
17C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
29C2C_2×\timesC2C_2 (1−6T+pT2)(1+2T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{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 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
47C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
53C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−12T+pT2)(1−8T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−2T+pT2)(1+14T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )
67C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
73C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1+62T2+p2T4 1 + 62 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 1−18T2+p2T4 1 - 18 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

−10.63205868806947687376151584522, −10.02903962845263956296816426871, −9.610325820479752915128161554525, −9.116461851488262454950794669344, −8.710983449597010051120350438492, −7.966346157839275655786998819682, −7.21349856853454183234034686305, −6.80685216687329560784577020357, −6.17347444353101165016767123962, −5.66940960198070682579828308583, −4.90472667772212871688171740074, −4.09321441703235896059660907164, −3.57735463172639924942421110349, −2.28237866716572804484448288641, −1.53239042869583881003940329862, 1.53239042869583881003940329862, 2.28237866716572804484448288641, 3.57735463172639924942421110349, 4.09321441703235896059660907164, 4.90472667772212871688171740074, 5.66940960198070682579828308583, 6.17347444353101165016767123962, 6.80685216687329560784577020357, 7.21349856853454183234034686305, 7.966346157839275655786998819682, 8.710983449597010051120350438492, 9.116461851488262454950794669344, 9.610325820479752915128161554525, 10.02903962845263956296816426871, 10.63205868806947687376151584522

Graph of the ZZ-function along the critical line