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Properties

Label 8-768e4-1.1-c1e4-0-13
Degree 88
Conductor 347892350976347892350976
Sign 11
Analytic cond. 1414.331414.33
Root an. cond. 2.476392.47639
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 + 6·9-s + 24·19-s − 4·25-s + 4·27-s + 8·43-s + 20·49-s − 96·57-s − 8·67-s + 24·73-s + 16·75-s − 37·81-s + 40·97-s + 28·121-s + 127-s − 32·129-s + 131-s + 137-s + 139-s − 80·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 144·171-s + ⋯
L(s)  = 1  − 2.30·3-s + 2·9-s + 5.50·19-s − 4/5·25-s + 0.769·27-s + 1.21·43-s + 20/7·49-s − 12.7·57-s − 0.977·67-s + 2.80·73-s + 1.84·75-s − 4.11·81-s + 4.06·97-s + 2.54·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.59·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 11.0·171-s + ⋯

Functional equation

Λ(s)=((232⋅34)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((232⋅34)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 232⋅342^{32} \cdot 3^{4}
Sign: 11
Analytic conductor: 1414.331414.33
Root analytic conductor: 2.476392.47639
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 232⋅34, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.7999864471.799986447
L(12)L(\frac12) ≈\approx 1.7999864471.799986447
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
3C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
good5C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
7C22C_2^2 (1−10T2+p2T4)2 ( 1 - 10 T^{2} + p^{2} T^{4} )^{2}
11C2C_2 (1−6T+pT2)2(1+6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2}
13C22C_2^2 (1−22T2+p2T4)2 ( 1 - 22 T^{2} + p^{2} T^{4} )^{2}
17C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
19C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
23C22C_2^2 (1+14T2+p2T4)2 ( 1 + 14 T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1+50T2+p2T4)2 ( 1 + 50 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1−58T2+p2T4)2 ( 1 - 58 T^{2} + p^{2} T^{4} )^{2}
37C22C_2^2 (1−38T2+p2T4)2 ( 1 - 38 T^{2} + p^{2} T^{4} )^{2}
41C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
43C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
47C22C_2^2 (1−34T2+p2T4)2 ( 1 - 34 T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1+34T2+p2T4)2 ( 1 + 34 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1−110T2+p2T4)2 ( 1 - 110 T^{2} + p^{2} T^{4} )^{2}
61C22C_2^2 (1−118T2+p2T4)2 ( 1 - 118 T^{2} + p^{2} T^{4} )^{2}
67C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
71C22C_2^2 (1+110T2+p2T4)2 ( 1 + 110 T^{2} + p^{2} T^{4} )^{2}
73C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
79C22C_2^2 (1+38T2+p2T4)2 ( 1 + 38 T^{2} + p^{2} T^{4} )^{2}
83C2C_2 (1−18T+pT2)2(1+18T+pT2)2 ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2}
89C22C_2^2 (1+110T2+p2T4)2 ( 1 + 110 T^{2} + p^{2} T^{4} )^{2}
97C2C_2 (1−10T+pT2)4 ( 1 - 10 T + 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

−7.44136780119340210157190534445, −7.21429759978967445603884307278, −6.98618684393061331716006404217, −6.74912287965975433785368990004, −6.41597717823481055078715679937, −5.99766610609118302943099009075, −5.84741892368229345307279839219, −5.78703279602356452623271551160, −5.71894897693456412260101745725, −5.27160400599861948464020251522, −5.09992699319263516767130241532, −5.01255937581447671062533828346, −4.76249596032280168370694187900, −4.42019268857796413045899048568, −4.00163574267424099965367794325, −3.78078906294554177843693138900, −3.37628091851646804423964913142, −3.15473104332175028005239317372, −2.97702873101470854607841532407, −2.60993133914374800312022424799, −2.04296572744955548990160430870, −1.71617496885459994640486993606, −0.898916531007710969705436753683, −0.831482585609022497589204033973, −0.74608280365055639237827980187, 0.74608280365055639237827980187, 0.831482585609022497589204033973, 0.898916531007710969705436753683, 1.71617496885459994640486993606, 2.04296572744955548990160430870, 2.60993133914374800312022424799, 2.97702873101470854607841532407, 3.15473104332175028005239317372, 3.37628091851646804423964913142, 3.78078906294554177843693138900, 4.00163574267424099965367794325, 4.42019268857796413045899048568, 4.76249596032280168370694187900, 5.01255937581447671062533828346, 5.09992699319263516767130241532, 5.27160400599861948464020251522, 5.71894897693456412260101745725, 5.78703279602356452623271551160, 5.84741892368229345307279839219, 5.99766610609118302943099009075, 6.41597717823481055078715679937, 6.74912287965975433785368990004, 6.98618684393061331716006404217, 7.21429759978967445603884307278, 7.44136780119340210157190534445

Graph of the ZZ-function along the critical line