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Properties

Label 8-3240e4-1.1-c1e4-0-6
Degree 88
Conductor 1.102×10141.102\times 10^{14}
Sign 11
Analytic cond. 448010.448010.
Root an. cond. 5.086405.08640
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  − 2·5-s − 7-s − 5·11-s − 5·13-s + 8·17-s + 4·19-s + 7·23-s + 25-s − 5·29-s − 5·31-s + 2·35-s − 12·37-s − 8·43-s − 5·47-s + 18·53-s + 10·55-s + 26·59-s − 2·61-s + 10·65-s + 14·67-s − 18·71-s + 5·77-s − 6·79-s − 10·83-s − 16·85-s + 6·89-s + 5·91-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 1.50·11-s − 1.38·13-s + 1.94·17-s + 0.917·19-s + 1.45·23-s + 1/5·25-s − 0.928·29-s − 0.898·31-s + 0.338·35-s − 1.97·37-s − 1.21·43-s − 0.729·47-s + 2.47·53-s + 1.34·55-s + 3.38·59-s − 0.256·61-s + 1.24·65-s + 1.71·67-s − 2.13·71-s + 0.569·77-s − 0.675·79-s − 1.09·83-s − 1.73·85-s + 0.635·89-s + 0.524·91-s + ⋯

Functional equation

Λ(s)=((212⋅316⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((212⋅316⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16} \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: 212⋅316⋅542^{12} \cdot 3^{16} \cdot 5^{4}
Sign: 11
Analytic conductor: 448010.448010.
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 212⋅316⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{12} \cdot 3^{16} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.7933637291.793363729
L(12)L(\frac12) ≈\approx 1.7933637291.793363729
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
5C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
good7D4×C2D_4\times C_2 1+T+T2−2pT3−8pT4−2p2T5+p2T6+p3T7+p4T8 1 + T + T^{2} - 2 p T^{3} - 8 p T^{4} - 2 p^{2} T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
11C2C_2×\timesC22C_2^2 (1+5T+pT2)2(1−5T+14T2−5pT3+p2T4) ( 1 + 5 T + p T^{2} )^{2}( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )
13D4×C2D_4\times C_2 1+5T+7T2−40T3−170T4−40pT5+7p2T6+5p3T7+p4T8 1 + 5 T + 7 T^{2} - 40 T^{3} - 170 T^{4} - 40 p T^{5} + 7 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
17C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
19C2C_2 (1−T+pT2)4 ( 1 - T + p T^{2} )^{4}
23D4×C2D_4\times C_2 1−7T+5T2+14T3+280T4+14pT5+5p2T6−7p3T7+p4T8 1 - 7 T + 5 T^{2} + 14 T^{3} + 280 T^{4} + 14 p T^{5} + 5 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1+5T−25T2−40T3+934T4−40pT5−25p2T6+5p3T7+p4T8 1 + 5 T - 25 T^{2} - 40 T^{3} + 934 T^{4} - 40 p T^{5} - 25 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
31D4×C2D_4\times C_2 1+5T−29T2−40T3+1180T4−40pT5−29p2T6+5p3T7+p4T8 1 + 5 T - 29 T^{2} - 40 T^{3} + 1180 T^{4} - 40 p T^{5} - 29 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
37D4D_{4} (1+6T+26T2+6pT3+p2T4)2 ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
41C23C_2^3 1−25T2−1056T4−25p2T6+p4T8 1 - 25 T^{2} - 1056 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8}
43C22C_2^2 (1+4T−27T2+4pT3+p2T4)2 ( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
47D4×C2D_4\times C_2 1+5T−61T2−40T3+4012T4−40pT5−61p2T6+5p3T7+p4T8 1 + 5 T - 61 T^{2} - 40 T^{3} + 4012 T^{4} - 40 p T^{5} - 61 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
53D4D_{4} (1−9T+112T2−9pT3+p2T4)2 ( 1 - 9 T + 112 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}
59C22C_2^2 (1−13T+110T2−13pT3+p2T4)2 ( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2}
61D4×C2D_4\times C_2 1+2T−62T2−112T3+391T4−112pT5−62p2T6+2p3T7+p4T8 1 + 2 T - 62 T^{2} - 112 T^{3} + 391 T^{4} - 112 p T^{5} - 62 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1−14T+70T2+112T3−1745T4+112pT5+70p2T6−14p3T7+p4T8 1 - 14 T + 70 T^{2} + 112 T^{3} - 1745 T^{4} + 112 p T^{5} + 70 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}
71D4D_{4} (1+9T+148T2+9pT3+p2T4)2 ( 1 + 9 T + 148 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2}
73C22C_2^2 (1−82T2+p2T4)2 ( 1 - 82 T^{2} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1+6T−74T2−288T3+3015T4−288pT5−74p2T6+6p3T7+p4T8 1 + 6 T - 74 T^{2} - 288 T^{3} + 3015 T^{4} - 288 p T^{5} - 74 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
83D4×C2D_4\times C_2 1+10T−34T2−320T3+2767T4−320pT5−34p2T6+10p3T7+p4T8 1 + 10 T - 34 T^{2} - 320 T^{3} + 2767 T^{4} - 320 p T^{5} - 34 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}
89D4D_{4} (1−3T+52T2−3pT3+p2T4)2 ( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
97C22C_2^2 (1+16T+159T2+16pT3+p2T4)2 ( 1 + 16 T + 159 T^{2} + 16 p T^{3} + 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

−6.09925083368134653594092433837, −5.63661864379034776542001094716, −5.44837942339292435963169058332, −5.43590690678611229897868269154, −5.42969412832855644202376512975, −5.28641907115237213210037992604, −5.06014006757891907360464634253, −4.88916031061894270373107541228, −4.31591032022742999178370831275, −4.18108384875981868390701580774, −4.13549531751586077961136025781, −3.93400762030274299023364035412, −3.45758623441384494762995693303, −3.45286782218502572716589536350, −2.97034135266453540496403006162, −2.96663573115802318698406795440, −2.93731378285775187473723937460, −2.63174143404559382185289653467, −2.06123706445758663165016355980, −1.92261902818342044740994526986, −1.79057216632657186770698716915, −1.23001109275533122997346350556, −0.988441127705432430257004561055, −0.39509856336968475628597315738, −0.39213207595486492653136790845, 0.39213207595486492653136790845, 0.39509856336968475628597315738, 0.988441127705432430257004561055, 1.23001109275533122997346350556, 1.79057216632657186770698716915, 1.92261902818342044740994526986, 2.06123706445758663165016355980, 2.63174143404559382185289653467, 2.93731378285775187473723937460, 2.96663573115802318698406795440, 2.97034135266453540496403006162, 3.45286782218502572716589536350, 3.45758623441384494762995693303, 3.93400762030274299023364035412, 4.13549531751586077961136025781, 4.18108384875981868390701580774, 4.31591032022742999178370831275, 4.88916031061894270373107541228, 5.06014006757891907360464634253, 5.28641907115237213210037992604, 5.42969412832855644202376512975, 5.43590690678611229897868269154, 5.44837942339292435963169058332, 5.63661864379034776542001094716, 6.09925083368134653594092433837

Graph of the ZZ-function along the critical line