↑

Properties

Label 8-1350e4-1.1-c1e4-0-4
Degree 88
Conductor 3.322×10123.322\times 10^{12}
Sign 11
Analytic cond. 13503.413503.4
Root an. cond. 3.283263.28326
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4-s + 7-s − 2·8-s − 3·11-s − 2·13-s + 2·14-s − 4·16-s − 18·17-s + 2·19-s − 6·22-s + 3·23-s − 4·26-s + 28-s − 3·29-s + 2·31-s − 2·32-s − 36·34-s + 16·37-s + 4·38-s − 6·41-s − 17·43-s − 3·44-s + 6·46-s + 9·47-s + 6·49-s − 2·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 0.377·7-s − 0.707·8-s − 0.904·11-s − 0.554·13-s + 0.534·14-s − 16-s − 4.36·17-s + 0.458·19-s − 1.27·22-s + 0.625·23-s − 0.784·26-s + 0.188·28-s − 0.557·29-s + 0.359·31-s − 0.353·32-s − 6.17·34-s + 2.63·37-s + 0.648·38-s − 0.937·41-s − 2.59·43-s − 0.452·44-s + 0.884·46-s + 1.31·47-s + 6/7·49-s − 0.277·52-s + ⋯

Functional equation

Λ(s)=((24⋅312⋅58)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((24⋅312⋅58)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \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⋅312⋅582^{4} \cdot 3^{12} \cdot 5^{8}
Sign: 11
Analytic conductor: 13503.413503.4
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅312⋅58, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{4} \cdot 3^{12} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.0801555993.080155599
L(12)L(\frac12) ≈\approx 3.0801555993.080155599
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}
3 1 1
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}
show more
show less
   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

−6.85939501111415035919907215913, −6.64360564819188993231464793411, −6.37221030211350448392634298960, −6.36858332515990500959934303678, −5.82101939346536012136074021965, −5.58035663198225969691658068112, −5.43063259331106849398368286666, −5.38505988644591281667851504292, −5.05287122772067412600783394698, −4.77247838521949474450934730247, −4.49309436647894845531686685883, −4.37766815554762989460098229166, −4.28606288454581634274843902220, −4.16250271313971930805068668779, −3.66027291201539329773769919083, −3.52963761548160609499957729765, −2.94958009748604298446120992469, −2.89505671910574810782854984780, −2.57623469464808744662191450288, −2.47099230915486505267427229671, −1.96516205169217768083173587473, −1.91976876368333965946803393861, −1.42986992319072475202883683175, −0.57768095327140122740756497340, −0.39325568901143939673284207071, 0.39325568901143939673284207071, 0.57768095327140122740756497340, 1.42986992319072475202883683175, 1.91976876368333965946803393861, 1.96516205169217768083173587473, 2.47099230915486505267427229671, 2.57623469464808744662191450288, 2.89505671910574810782854984780, 2.94958009748604298446120992469, 3.52963761548160609499957729765, 3.66027291201539329773769919083, 4.16250271313971930805068668779, 4.28606288454581634274843902220, 4.37766815554762989460098229166, 4.49309436647894845531686685883, 4.77247838521949474450934730247, 5.05287122772067412600783394698, 5.38505988644591281667851504292, 5.43063259331106849398368286666, 5.58035663198225969691658068112, 5.82101939346536012136074021965, 6.36858332515990500959934303678, 6.37221030211350448392634298960, 6.64360564819188993231464793411, 6.85939501111415035919907215913

Graph of the ZZ-function along the critical line