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Properties

Label 8-4800e4-1.1-c1e4-0-10
Degree 88
Conductor 5.308×10145.308\times 10^{14}
Sign 11
Analytic cond. 2.15810×1062.15810\times 10^{6}
Root an. cond. 6.190976.19097
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 10·9-s + 20·27-s − 24·41-s − 16·43-s + 4·49-s + 16·67-s + 35·81-s − 24·89-s − 48·107-s + 44·121-s − 96·123-s + 127-s − 64·129-s + 131-s + 137-s + 139-s + 16·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2.30·3-s + 10/3·9-s + 3.84·27-s − 3.74·41-s − 2.43·43-s + 4/7·49-s + 1.95·67-s + 35/9·81-s − 2.54·89-s − 4.64·107-s + 4·121-s − 8.65·123-s + 0.0887·127-s − 5.63·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=((224⋅34⋅58)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅34⋅58)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅34⋅582^{24} \cdot 3^{4} \cdot 5^{8}
Sign: 11
Analytic conductor: 2.15810×1062.15810\times 10^{6}
Root analytic conductor: 6.190976.19097
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅34⋅58, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 9.5691771509.569177150
L(12)L(\frac12) ≈\approx 9.5691771509.569177150
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 (1−T)4 ( 1 - T )^{4}
5 1 1
good7C2C_2 (1−4T+pT2)2(1+4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}
11C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
13C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
17C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
19C22C_2^2 (1−22T2+p2T4)2 ( 1 - 22 T^{2} + p^{2} T^{4} )^{2}
23C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1−46T2+p2T4)2 ( 1 - 46 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+50T2+p2T4)2 ( 1 + 50 T^{2} + p^{2} T^{4} )^{2}
37C22C_2^2 (1+26T2+p2T4)2 ( 1 + 26 T^{2} + p^{2} T^{4} )^{2}
41C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
43C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
47C22C_2^2 (1−46T2+p2T4)2 ( 1 - 46 T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1+94T2+p2T4)2 ( 1 + 94 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+26T2+p2T4)2 ( 1 + 26 T^{2} + p^{2} T^{4} )^{2}
61C2C_2 (1−14T+pT2)2(1+14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2}
67C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
71C22C_2^2 (1+94T2+p2T4)2 ( 1 + 94 T^{2} + p^{2} T^{4} )^{2}
73C22C_2^2 (1−142T2+p2T4)2 ( 1 - 142 T^{2} + p^{2} T^{4} )^{2}
79C22C_2^2 (1+50T2+p2T4)2 ( 1 + 50 T^{2} + p^{2} T^{4} )^{2}
83C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
89C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
97C22C_2^2 (1−190T2+p2T4)2 ( 1 - 190 T^{2} + p^{2} T^{4} )^{2}
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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

−5.84437107999515757029604054952, −5.48919479527498536386684897525, −5.47322983344339499908804413106, −5.09335812957088570680207045079, −4.95684601069507077375726667596, −4.85307968993678464529796509549, −4.82014774512490675407827899861, −4.31470727473056535141488689299, −4.20687837121113626552366424951, −3.82863118148668849192396788582, −3.82781902472090237279117887154, −3.57534614438738266602058295039, −3.45769240653537818604853541824, −3.26545991257441737057795762031, −3.00783085860334664774361951876, −2.66804575418411085169993260402, −2.60416910103091292378355961539, −2.34627932300160117978516015643, −2.23029950200667800324284693604, −1.62116629901489133674584384648, −1.57373969030464851546952209057, −1.49579243033920072884898908842, −1.31777832331629305834978479560, −0.50469519054015456670412738119, −0.36414243055218626506443927057, 0.36414243055218626506443927057, 0.50469519054015456670412738119, 1.31777832331629305834978479560, 1.49579243033920072884898908842, 1.57373969030464851546952209057, 1.62116629901489133674584384648, 2.23029950200667800324284693604, 2.34627932300160117978516015643, 2.60416910103091292378355961539, 2.66804575418411085169993260402, 3.00783085860334664774361951876, 3.26545991257441737057795762031, 3.45769240653537818604853541824, 3.57534614438738266602058295039, 3.82781902472090237279117887154, 3.82863118148668849192396788582, 4.20687837121113626552366424951, 4.31470727473056535141488689299, 4.82014774512490675407827899861, 4.85307968993678464529796509549, 4.95684601069507077375726667596, 5.09335812957088570680207045079, 5.47322983344339499908804413106, 5.48919479527498536386684897525, 5.84437107999515757029604054952

Graph of the ZZ-function along the critical line