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Properties

Label 8-160e4-1.1-c3e4-0-0
Degree 88
Conductor 655360000655360000
Sign 11
Analytic cond. 7942.267942.26
Root an. cond. 3.072503.07250
Motivic weight 33
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  + 250·25-s − 1.22e3·29-s − 478·81-s + 5.54e3·89-s − 1.51e3·101-s − 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 2·25-s − 7.83·29-s − 0.655·81-s + 6.60·89-s − 1.48·101-s − 4·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + ⋯

Functional equation

Λ(s)=((220⋅54)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((220⋅54)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅542^{20} \cdot 5^{4}
Sign: 11
Analytic conductor: 7942.267942.26
Root analytic conductor: 3.072503.07250
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅54, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{20} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.50885856550.5088585655
L(12)L(\frac12) ≈\approx 0.50885856550.5088585655
L(52)L(\frac{5}{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−p3T2)2 ( 1 - p^{3} T^{2} )^{2}
good3C22C_2^2×\timesC22C_2^2 (1−14T+98T2−14p3T3+p6T4)(1+14T+98T2+14p3T3+p6T4) ( 1 - 14 T + 98 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} )
7C22C_2^2×\timesC22C_2^2 (1−18T+162T2−18p3T3+p6T4)(1+18T+162T2+18p3T3+p6T4) ( 1 - 18 T + 162 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} )
11C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
13C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
17C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
19C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
23C22C_2^2×\timesC22C_2^2 (1−134T+8978T2−134p3T3+p6T4)(1+134T+8978T2+134p3T3+p6T4) ( 1 - 134 T + 8978 T^{2} - 134 p^{3} T^{3} + p^{6} T^{4} )( 1 + 134 T + 8978 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} )
29C2C_2 (1+306T+p3T2)4 ( 1 + 306 T + p^{3} T^{2} )^{4}
31C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
37C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
41C22C_2^2 (1−74338T2+p6T4)2 ( 1 - 74338 T^{2} + p^{6} T^{4} )^{2}
43C22C_2^2×\timesC22C_2^2 (1−594T+176418T2−594p3T3+p6T4)(1+594T+176418T2+594p3T3+p6T4) ( 1 - 594 T + 176418 T^{2} - 594 p^{3} T^{3} + p^{6} T^{4} )( 1 + 594 T + 176418 T^{2} + 594 p^{3} T^{3} + p^{6} T^{4} )
47C22C_2^2×\timesC22C_2^2 (1−602T+181202T2−602p3T3+p6T4)(1+602T+181202T2+602p3T3+p6T4) ( 1 - 602 T + 181202 T^{2} - 602 p^{3} T^{3} + p^{6} T^{4} )( 1 + 602 T + 181202 T^{2} + 602 p^{3} T^{3} + p^{6} T^{4} )
53C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
59C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
61C22C_2^2 (1+452342T2+p6T4)2 ( 1 + 452342 T^{2} + p^{6} T^{4} )^{2}
67C22C_2^2×\timesC22C_2^2 (1−1098T+602802T2−1098p3T3+p6T4)(1+1098T+602802T2+1098p3T3+p6T4) ( 1 - 1098 T + 602802 T^{2} - 1098 p^{3} T^{3} + p^{6} T^{4} )( 1 + 1098 T + 602802 T^{2} + 1098 p^{3} T^{3} + p^{6} T^{4} )
71C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
73C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
79C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
83C22C_2^2×\timesC22C_2^2 (1−154T+11858T2−154p3T3+p6T4)(1+154T+11858T2+154p3T3+p6T4) ( 1 - 154 T + 11858 T^{2} - 154 p^{3} T^{3} + p^{6} T^{4} )( 1 + 154 T + 11858 T^{2} + 154 p^{3} T^{3} + p^{6} T^{4} )
89C2C_2 (1−1386T+p3T2)4 ( 1 - 1386 T + p^{3} T^{2} )^{4}
97C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
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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

−9.142440889879378783652296202863, −9.084321433929143711189704304566, −8.296471432672225401798581744118, −8.121616736501044980628185671036, −7.76447838998563013933559703178, −7.52858319916217016773528016669, −7.42208168823084292373623505920, −6.98578195471648706148523551539, −6.86068672533443588908611754775, −6.38507898824676997381134157602, −5.90699923915792669497212227410, −5.89157886146161490282339699735, −5.35982987533224370712558748595, −5.13166426923616119496647256907, −5.10593770879553400022617563211, −4.32361141496976920208105231329, −4.05964249130542937344421446193, −3.59059949057182386486378014440, −3.55265180026937262639531173105, −3.08780382429683217382433433786, −2.36667406218455127757591975495, −1.93666941953532595376525967970, −1.78298412618732499995550685063, −0.997949833365966833715204881377, −0.16759679735773697081094993538, 0.16759679735773697081094993538, 0.997949833365966833715204881377, 1.78298412618732499995550685063, 1.93666941953532595376525967970, 2.36667406218455127757591975495, 3.08780382429683217382433433786, 3.55265180026937262639531173105, 3.59059949057182386486378014440, 4.05964249130542937344421446193, 4.32361141496976920208105231329, 5.10593770879553400022617563211, 5.13166426923616119496647256907, 5.35982987533224370712558748595, 5.89157886146161490282339699735, 5.90699923915792669497212227410, 6.38507898824676997381134157602, 6.86068672533443588908611754775, 6.98578195471648706148523551539, 7.42208168823084292373623505920, 7.52858319916217016773528016669, 7.76447838998563013933559703178, 8.121616736501044980628185671036, 8.296471432672225401798581744118, 9.084321433929143711189704304566, 9.142440889879378783652296202863

Graph of the ZZ-function along the critical line