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Properties

Label 8-2160e4-1.1-c1e4-0-7
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 88495.988495.9
Root an. cond. 4.153034.15303
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·11-s + 8·19-s + 5·25-s + 14·29-s − 12·31-s + 10·41-s − 13·49-s + 8·55-s − 24·59-s + 14·61-s − 40·71-s − 8·79-s − 60·89-s − 16·95-s + 36·101-s + 36·109-s + 26·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s − 28·145-s + 149-s + 151-s + 24·155-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.20·11-s + 1.83·19-s + 25-s + 2.59·29-s − 2.15·31-s + 1.56·41-s − 1.85·49-s + 1.07·55-s − 3.12·59-s + 1.79·61-s − 4.74·71-s − 0.900·79-s − 6.35·89-s − 1.64·95-s + 3.58·101-s + 3.44·109-s + 2.36·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.32·145-s + 0.0819·149-s + 0.0813·151-s + 1.92·155-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅312⋅542^{16} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 88495.988495.9
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.78358192830.7835819283
L(12)L(\frac12) ≈\approx 0.78358192830.7835819283
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
3 1 1
5C22C_2^2 1+2T−T2+2pT3+p2T4 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4}
good7C22C_2^2×\timesC22C_2^2 (1+2T2+p2T4)(1+11T2+p2T4) ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} )
11C22C_2^2 (1+2T−7T2+2pT3+p2T4)2 ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
13C22C_2^2×\timesC22C_2^2 (1−T2+p2T4)(1+23T2+p2T4) ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} )
17C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
19C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
23C23C_2^3 1+45T2+1496T4+45p2T6+p4T8 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1−7T+20T2−7pT3+p2T4)2 ( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+6T+5T2+6pT3+p2T4)2 ( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
37C2C_2 (1−12T+pT2)2(1+12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}
41C22C_2^2 (1−5T−16T2−5pT3+p2T4)2 ( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}
43C23C_2^3 1−58T2+1515T4−58p2T6+p4T8 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8}
47C23C_2^3 1+13T2−2040T4+13p2T6+p4T8 1 + 13 T^{2} - 2040 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8}
53C22C_2^2 (1−42T2+p2T4)2 ( 1 - 42 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+12T+85T2+12pT3+p2T4)2 ( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
61C22C_2^2 (1−7T−12T2−7pT3+p2T4)2 ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}
67C22C_2^2×\timesC22C_2^2 (1−13T2+p2T4)(1+122T2+p2T4) ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} )
71C2C_2 (1+10T+pT2)4 ( 1 + 10 T + p T^{2} )^{4}
73C22C_2^2 (1−130T2+p2T4)2 ( 1 - 130 T^{2} + p^{2} T^{4} )^{2}
79C2C_2 (1−13T+pT2)2(1+17T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2}
83C23C_2^3 1+141T2+12992T4+141p2T6+p4T8 1 + 141 T^{2} + 12992 T^{4} + 141 p^{2} T^{6} + p^{4} T^{8}
89C2C_2 (1+15T+pT2)4 ( 1 + 15 T + p T^{2} )^{4}
97C23C_2^3 1−62T2−5565T4−62p2T6+p4T8 1 - 62 T^{2} - 5565 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.51705240166213152007570652566, −6.34084420252208200709776897714, −5.88839803251556337528240864825, −5.69255090719242987571132975412, −5.61486154709628952937810949744, −5.55741756099638326030186871949, −5.21595294550443145040599821086, −4.87665054025083822360361555737, −4.59672595427736479484994834085, −4.59132994242869465085694220618, −4.32970464568374621884797804744, −4.31471640458441348609211874951, −3.85805029406410296845859215083, −3.38112363135966990741075046991, −3.33287295139578123854568996966, −3.10192491135609643412026370188, −2.93147751856421482378889068410, −2.80542572569371836071911928753, −2.48806038583656296070306932190, −2.00374971495860340207095191435, −1.73697515878946547055340031066, −1.31427732674723225173679902676, −1.21524416204542708796084851731, −0.66752585687098120215768822554, −0.18660954820541349242208180974, 0.18660954820541349242208180974, 0.66752585687098120215768822554, 1.21524416204542708796084851731, 1.31427732674723225173679902676, 1.73697515878946547055340031066, 2.00374971495860340207095191435, 2.48806038583656296070306932190, 2.80542572569371836071911928753, 2.93147751856421482378889068410, 3.10192491135609643412026370188, 3.33287295139578123854568996966, 3.38112363135966990741075046991, 3.85805029406410296845859215083, 4.31471640458441348609211874951, 4.32970464568374621884797804744, 4.59132994242869465085694220618, 4.59672595427736479484994834085, 4.87665054025083822360361555737, 5.21595294550443145040599821086, 5.55741756099638326030186871949, 5.61486154709628952937810949744, 5.69255090719242987571132975412, 5.88839803251556337528240864825, 6.34084420252208200709776897714, 6.51705240166213152007570652566

Graph of the ZZ-function along the critical line