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Properties

Label 8-160e4-1.1-c1e4-0-1
Degree 88
Conductor 655360000655360000
Sign 11
Analytic cond. 2.664332.66433
Root an. cond. 1.130311.13031
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  + 10·25-s + 24·29-s + 2·81-s − 24·89-s − 72·101-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-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 + 4.45·29-s + 2/9·81-s − 2.54·89-s − 7.16·101-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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 + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

Λ(s)=((220⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((220⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \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: 220⋅542^{20} \cdot 5^{4}
Sign: 11
Analytic conductor: 2.664332.66433
Root analytic conductor: 1.130311.13031
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{20} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.4577807101.457780710
L(12)L(\frac12) ≈\approx 1.4577807101.457780710
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−pT2)2 ( 1 - p T^{2} )^{2}
good3C22C_2^2×\timesC22C_2^2 (1−2T+2T2−2pT3+p2T4)(1+2T+2T2+2pT3+p2T4) ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )
7C22C_2^2×\timesC22C_2^2 (1−6T+18T2−6pT3+p2T4)(1+6T+18T2+6pT3+p2T4) ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )
11C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
13C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
17C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
19C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
23C22C_2^2×\timesC22C_2^2 (1−2T+2T2−2pT3+p2T4)(1+2T+2T2+2pT3+p2T4) ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )
29C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
31C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
37C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
41C22C_2^2 (1+62T2+p2T4)2 ( 1 + 62 T^{2} + p^{2} T^{4} )^{2}
43C22C_2^2×\timesC22C_2^2 (1−18T+162T2−18pT3+p2T4)(1+18T+162T2+18pT3+p2T4) ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} )
47C22C_2^2×\timesC22C_2^2 (1−14T+98T2−14pT3+p2T4)(1+14T+98T2+14pT3+p2T4) ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )
53C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
59C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
61C22C_2^2 (1−58T2+p2T4)2 ( 1 - 58 T^{2} + p^{2} T^{4} )^{2}
67C22C_2^2×\timesC22C_2^2 (1−6T+18T2−6pT3+p2T4)(1+6T+18T2+6pT3+p2T4) ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )
71C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
73C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
79C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
83C22C_2^2×\timesC22C_2^2 (1−22T+242T2−22pT3+p2T4)(1+22T+242T2+22pT3+p2T4) ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} )
89C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
97C2C_2 (1−pT2)4 ( 1 - p 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

−9.376028053791290275288534376888, −9.300931496355539349042987080767, −8.931235541128439524930930999071, −8.439039713800605599546754603685, −8.348622798880760706844654204351, −8.290531743834660130877881971853, −8.023666956296543312925465170693, −7.42541760646773038739521629905, −7.24428262020010073229159694421, −6.74347725118726718629208840651, −6.65989392962173684406985324141, −6.43519184526866561268525595499, −6.31002618409748606745822405705, −5.43352400992124616358714228380, −5.33588751391846393555097535154, −5.27231567091486983092416864152, −4.58001151796818634463342727799, −4.32262036432335171751972215288, −4.24641408342865867467795153238, −3.60042315231287594743817683746, −2.86185531446723818518215656358, −2.75334185645567540086841645499, −2.71110513399110985174563244987, −1.49656336476859230117027825145, −1.05253035425592649295608201792, 1.05253035425592649295608201792, 1.49656336476859230117027825145, 2.71110513399110985174563244987, 2.75334185645567540086841645499, 2.86185531446723818518215656358, 3.60042315231287594743817683746, 4.24641408342865867467795153238, 4.32262036432335171751972215288, 4.58001151796818634463342727799, 5.27231567091486983092416864152, 5.33588751391846393555097535154, 5.43352400992124616358714228380, 6.31002618409748606745822405705, 6.43519184526866561268525595499, 6.65989392962173684406985324141, 6.74347725118726718629208840651, 7.24428262020010073229159694421, 7.42541760646773038739521629905, 8.023666956296543312925465170693, 8.290531743834660130877881971853, 8.348622798880760706844654204351, 8.439039713800605599546754603685, 8.931235541128439524930930999071, 9.300931496355539349042987080767, 9.376028053791290275288534376888

Graph of the ZZ-function along the critical line