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Properties

Label 8-450e4-1.1-c1e4-0-9
Degree 88
Conductor 4100625000041006250000
Sign 11
Analytic cond. 166.708166.708
Root an. cond. 1.895591.89559
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  − 2·2-s − 2·3-s + 4-s + 4·6-s + 7-s + 2·8-s − 3·9-s + 3·11-s − 2·12-s − 2·13-s − 2·14-s − 4·16-s + 18·17-s + 6·18-s + 2·19-s − 2·21-s − 6·22-s − 3·23-s − 4·24-s + 4·26-s + 14·27-s + 28-s + 3·29-s + 2·31-s + 2·32-s − 6·33-s − 36·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s + 0.377·7-s + 0.707·8-s − 9-s + 0.904·11-s − 0.577·12-s − 0.554·13-s − 0.534·14-s − 16-s + 4.36·17-s + 1.41·18-s + 0.458·19-s − 0.436·21-s − 1.27·22-s − 0.625·23-s − 0.816·24-s + 0.784·26-s + 2.69·27-s + 0.188·28-s + 0.557·29-s + 0.359·31-s + 0.353·32-s − 1.04·33-s − 6.17·34-s + ⋯

Functional equation

Λ(s)=((24⋅38⋅58)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((24⋅38⋅58)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \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: 24⋅38⋅582^{4} \cdot 3^{8} \cdot 5^{8}
Sign: 11
Analytic conductor: 166.708166.708
Root analytic conductor: 1.895591.89559
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅38⋅58, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{4} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.75132769360.7513276936
L(12)L(\frac12) ≈\approx 0.75132769360.7513276936
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)
bad2C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
3C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
5 1 1
good7D4×C2D_4\times C_2 1−T−5T2+8T3−20T4+8pT5−5p2T6−p3T7+p4T8 1 - T - 5 T^{2} + 8 T^{3} - 20 T^{4} + 8 p T^{5} - 5 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1−3T−7T2+18T3+36T4+18pT5−7p2T6−3p3T7+p4T8 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
13D4×C2D_4\times C_2 1+2T+10T2−64T3−185T4−64pT5+10p2T6+2p3T7+p4T8 1 + 2 T + 10 T^{2} - 64 T^{3} - 185 T^{4} - 64 p T^{5} + 10 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
17D4D_{4} (1−9T+46T2−9pT3+p2T4)2 ( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}
19D4D_{4} (1−T+30T2−pT3+p2T4)2 ( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} )^{2}
23D4×C2D_4\times C_2 1+3T−31T2−18T3+864T4−18pT5−31p2T6+3p3T7+p4T8 1 + 3 T - 31 T^{2} - 18 T^{3} + 864 T^{4} - 18 p T^{5} - 31 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1−3T−43T2+18T3+1602T4+18pT5−43p2T6−3p3T7+p4T8 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
31D4×C2D_4\times C_2 1−2T−26T2+64T3−185T4+64pT5−26p2T6−2p3T7+p4T8 1 - 2 T - 26 T^{2} + 64 T^{3} - 185 T^{4} + 64 p T^{5} - 26 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
37C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
41C22C_2^2 (1−3T−32T2−3pT3+p2T4)2 ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1+17T+139T2+1088T3+8224T4+1088pT5+139p2T6+17p3T7+p4T8 1 + 17 T + 139 T^{2} + 1088 T^{3} + 8224 T^{4} + 1088 p T^{5} + 139 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1+9T−25T2+108T3+5220T4+108pT5−25p2T6+9p3T7+p4T8 1 + 9 T - 25 T^{2} + 108 T^{3} + 5220 T^{4} + 108 p T^{5} - 25 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}
53C22C_2^2 (1−26T2+p2T4)2 ( 1 - 26 T^{2} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1+3T−103T2−18T3+8532T4−18pT5−103p2T6+3p3T7+p4T8 1 + 3 T - 103 T^{2} - 18 T^{3} + 8532 T^{4} - 18 p T^{5} - 103 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1+T−47T2−74T3−1478T4−74pT5−47p2T6+p3T7+p4T8 1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 74 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
67C22C_2^2 (1+7T−18T2+7pT3+p2T4)2 ( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2}
71C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
73D4D_{4} (1−11T+102T2−11pT3+p2T4)2 ( 1 - 11 T + 102 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2}
79C22C_2^2 (1+2T−75T2+2pT3+p2T4)2 ( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−9T−97T2−108T3+18072T4−108pT5−97p2T6−9p3T7+p4T8 1 - 9 T - 97 T^{2} - 108 T^{3} + 18072 T^{4} - 108 p T^{5} - 97 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}
89D4D_{4} (1−15T+160T2−15pT3+p2T4)2 ( 1 - 15 T + 160 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1+11T−95T2+242T3+25510T4+242pT5−95p2T6+11p3T7+p4T8 1 + 11 T - 95 T^{2} + 242 T^{3} + 25510 T^{4} + 242 p T^{5} - 95 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}
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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

−8.238907163307847926973934599389, −7.59642553936461887459791423463, −7.58181438181280895180891645024, −7.56168929819908919274144386886, −7.39557845651719156679207722853, −6.72138483076677685519083495070, −6.45727953836589963641012956699, −6.24843277043642241968095167081, −6.01243059635382557420088787259, −5.97225882102372490357664095155, −5.37226378574877034614356017304, −5.36962303761258296357876959444, −5.14529696431151940772261773175, −4.80381608457137654242643483043, −4.39829875938970653391149644738, −4.32873757139216511598232275913, −3.66282152371905287077999733307, −3.26973320707507731294267519279, −3.21436384597287508819489242448, −2.94340361932433208035035069957, −2.37434490956744227691583248713, −1.74274670551797321247400523076, −1.37399023293577192992577820503, −0.795769076564163916437065553995, −0.69330471787648547657972885818, 0.69330471787648547657972885818, 0.795769076564163916437065553995, 1.37399023293577192992577820503, 1.74274670551797321247400523076, 2.37434490956744227691583248713, 2.94340361932433208035035069957, 3.21436384597287508819489242448, 3.26973320707507731294267519279, 3.66282152371905287077999733307, 4.32873757139216511598232275913, 4.39829875938970653391149644738, 4.80381608457137654242643483043, 5.14529696431151940772261773175, 5.36962303761258296357876959444, 5.37226378574877034614356017304, 5.97225882102372490357664095155, 6.01243059635382557420088787259, 6.24843277043642241968095167081, 6.45727953836589963641012956699, 6.72138483076677685519083495070, 7.39557845651719156679207722853, 7.56168929819908919274144386886, 7.58181438181280895180891645024, 7.59642553936461887459791423463, 8.238907163307847926973934599389

Graph of the ZZ-function along the critical line