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Properties

Label 8-1449e4-1.1-c1e4-0-5
Degree 88
Conductor 4.408×10124.408\times 10^{12}
Sign 11
Analytic cond. 17921.817921.8
Root an. cond. 3.401513.40151
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4-s − 2·5-s + 4·7-s + 2·8-s + 4·10-s − 8·11-s − 2·13-s − 8·14-s + 8·16-s − 8·19-s + 2·20-s + 16·22-s + 4·23-s − 11·25-s + 4·26-s − 4·28-s − 8·29-s + 4·31-s − 10·32-s − 8·35-s − 4·37-s + 16·38-s − 4·40-s − 12·41-s − 2·43-s + 8·44-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s − 0.894·5-s + 1.51·7-s + 0.707·8-s + 1.26·10-s − 2.41·11-s − 0.554·13-s − 2.13·14-s + 2·16-s − 1.83·19-s + 0.447·20-s + 3.41·22-s + 0.834·23-s − 2.19·25-s + 0.784·26-s − 0.755·28-s − 1.48·29-s + 0.718·31-s − 1.76·32-s − 1.35·35-s − 0.657·37-s + 2.59·38-s − 0.632·40-s − 1.87·41-s − 0.304·43-s + 1.20·44-s + ⋯

Functional equation

Λ(s)=((38⋅74⋅234)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((38⋅74⋅234)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 38⋅74⋅2343^{8} \cdot 7^{4} \cdot 23^{4}
Sign: 11
Analytic conductor: 17921.817921.8
Root analytic conductor: 3.401513.40151
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 38⋅74⋅234, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{8} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
7C1C_1 (1−T)4 ( 1 - T )^{4}
23C1C_1 (1−T)4 ( 1 - T )^{4}
good2C2≀C2≀C2C_2 \wr C_2\wr C_2 1+pT+5T2+5pT3+13T4+5p2T5+5p2T6+p4T7+p4T8 1 + p T + 5 T^{2} + 5 p T^{3} + 13 T^{4} + 5 p^{2} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8}
5C2≀C2≀C2C_2 \wr C_2\wr C_2 1+2T+3pT2+24T3+101T4+24pT5+3p3T6+2p3T7+p4T8 1 + 2 T + 3 p T^{2} + 24 T^{3} + 101 T^{4} + 24 p T^{5} + 3 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
11C2≀C2≀C2C_2 \wr C_2\wr C_2 1+8T+52T2+20pT3+829T4+20p2T5+52p2T6+8p3T7+p4T8 1 + 8 T + 52 T^{2} + 20 p T^{3} + 829 T^{4} + 20 p^{2} T^{5} + 52 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
13C2≀C2≀C2C_2 \wr C_2\wr C_2 1+2T+17T2−8T3+55T4−8pT5+17p2T6+2p3T7+p4T8 1 + 2 T + 17 T^{2} - 8 T^{3} + 55 T^{4} - 8 p T^{5} + 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
17C2≀C2≀C2C_2 \wr C_2\wr C_2 1+50T2−16T3+67pT4−16pT5+50p2T6+p4T8 1 + 50 T^{2} - 16 T^{3} + 67 p T^{4} - 16 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8}
19C2≀C2≀C2C_2 \wr C_2\wr C_2 1+8T+70T2+336T3+1763T4+336pT5+70p2T6+8p3T7+p4T8 1 + 8 T + 70 T^{2} + 336 T^{3} + 1763 T^{4} + 336 p T^{5} + 70 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
29C2≀C2≀C2C_2 \wr C_2\wr C_2 1+8T+110T2+640T3+4651T4+640pT5+110p2T6+8p3T7+p4T8 1 + 8 T + 110 T^{2} + 640 T^{3} + 4651 T^{4} + 640 p T^{5} + 110 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
31C2≀C2≀C2C_2 \wr C_2\wr C_2 1−4T+52T2−420T3+1421T4−420pT5+52p2T6−4p3T7+p4T8 1 - 4 T + 52 T^{2} - 420 T^{3} + 1421 T^{4} - 420 p T^{5} + 52 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
37C2≀C2≀C2C_2 \wr C_2\wr C_2 1+4T+20T2−180T3−1755T4−180pT5+20p2T6+4p3T7+p4T8 1 + 4 T + 20 T^{2} - 180 T^{3} - 1755 T^{4} - 180 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
41C2≀C2≀C2C_2 \wr C_2\wr C_2 1+12T+84T2+604T3+4757T4+604pT5+84p2T6+12p3T7+p4T8 1 + 12 T + 84 T^{2} + 604 T^{3} + 4757 T^{4} + 604 p T^{5} + 84 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
43C2≀C2≀C2C_2 \wr C_2\wr C_2 1+2T+131T2+264T3+7791T4+264pT5+131p2T6+2p3T7+p4T8 1 + 2 T + 131 T^{2} + 264 T^{3} + 7791 T^{4} + 264 p T^{5} + 131 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
47C2≀C2≀C2C_2 \wr C_2\wr C_2 1+16T+156T2+1232T3+9222T4+1232pT5+156p2T6+16p3T7+p4T8 1 + 16 T + 156 T^{2} + 1232 T^{3} + 9222 T^{4} + 1232 p T^{5} + 156 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
53C2≀C2≀C2C_2 \wr C_2\wr C_2 1−2T+135T2−272T3+9899T4−272pT5+135p2T6−2p3T7+p4T8 1 - 2 T + 135 T^{2} - 272 T^{3} + 9899 T^{4} - 272 p T^{5} + 135 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
59C2≀C2≀C2C_2 \wr C_2\wr C_2 1+22T+237T2+1544T3+9747T4+1544pT5+237p2T6+22p3T7+p4T8 1 + 22 T + 237 T^{2} + 1544 T^{3} + 9747 T^{4} + 1544 p T^{5} + 237 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8}
61C2≀C2≀C2C_2 \wr C_2\wr C_2 1−22T+333T2−3320T3+29001T4−3320pT5+333p2T6−22p3T7+p4T8 1 - 22 T + 333 T^{2} - 3320 T^{3} + 29001 T^{4} - 3320 p T^{5} + 333 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8}
67C2≀C2≀C2C_2 \wr C_2\wr C_2 1+10T+225T2+1796T3+21059T4+1796pT5+225p2T6+10p3T7+p4T8 1 + 10 T + 225 T^{2} + 1796 T^{3} + 21059 T^{4} + 1796 p T^{5} + 225 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}
71C2≀C2≀C2C_2 \wr C_2\wr C_2 1+46T+1029T2+14628T3+145493T4+14628pT5+1029p2T6+46p3T7+p4T8 1 + 46 T + 1029 T^{2} + 14628 T^{3} + 145493 T^{4} + 14628 p T^{5} + 1029 p^{2} T^{6} + 46 p^{3} T^{7} + p^{4} T^{8}
73C2≀C2≀C2C_2 \wr C_2\wr C_2 1+16T+316T2+3284T3+35053T4+3284pT5+316p2T6+16p3T7+p4T8 1 + 16 T + 316 T^{2} + 3284 T^{3} + 35053 T^{4} + 3284 p T^{5} + 316 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
79C2≀C2≀C2C_2 \wr C_2\wr C_2 1−16T+196T2−1460T3+15037T4−1460pT5+196p2T6−16p3T7+p4T8 1 - 16 T + 196 T^{2} - 1460 T^{3} + 15037 T^{4} - 1460 p T^{5} + 196 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
83C2≀C2≀C2C_2 \wr C_2\wr C_2 1−4T+90T2+1032T3−2269T4+1032pT5+90p2T6−4p3T7+p4T8 1 - 4 T + 90 T^{2} + 1032 T^{3} - 2269 T^{4} + 1032 p T^{5} + 90 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
89C2≀C2≀C2C_2 \wr C_2\wr C_2 1+34T+671T2+9272T3+99277T4+9272pT5+671p2T6+34p3T7+p4T8 1 + 34 T + 671 T^{2} + 9272 T^{3} + 99277 T^{4} + 9272 p T^{5} + 671 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8}
97C2≀C2≀C2C_2 \wr C_2\wr C_2 1+8T+110T2−1408T3−9381T4−1408pT5+110p2T6+8p3T7+p4T8 1 + 8 T + 110 T^{2} - 1408 T^{3} - 9381 T^{4} - 1408 p T^{5} + 110 p^{2} T^{6} + 8 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

−7.51039557222115745086939640217, −7.19768271770094931055091007013, −6.89530605928380126533230829220, −6.71564651288330159033505106747, −6.38348899878331608097486707110, −6.08267645739920930578407308236, −5.83098624788739545112964869548, −5.55390386843500072023791192832, −5.54923108233996494949466598593, −5.31248194038059422200757493095, −4.95350728577447142796534848196, −4.89520264559482491669912567369, −4.56893867806062625997458280930, −4.28845297771897567160803456640, −4.18254929515750532470716379538, −3.97156600183317253188879815063, −3.53434126585843400620190006400, −3.28622703884232841217949607852, −3.06198789609993881235584252469, −2.77367237532757570553731331136, −2.39953716149398114765788171161, −2.24043287633578655260408926465, −1.62846200567657853501367336518, −1.47786163702019738409619452006, −1.32072307313678524800888992946, 0, 0, 0, 0, 1.32072307313678524800888992946, 1.47786163702019738409619452006, 1.62846200567657853501367336518, 2.24043287633578655260408926465, 2.39953716149398114765788171161, 2.77367237532757570553731331136, 3.06198789609993881235584252469, 3.28622703884232841217949607852, 3.53434126585843400620190006400, 3.97156600183317253188879815063, 4.18254929515750532470716379538, 4.28845297771897567160803456640, 4.56893867806062625997458280930, 4.89520264559482491669912567369, 4.95350728577447142796534848196, 5.31248194038059422200757493095, 5.54923108233996494949466598593, 5.55390386843500072023791192832, 5.83098624788739545112964869548, 6.08267645739920930578407308236, 6.38348899878331608097486707110, 6.71564651288330159033505106747, 6.89530605928380126533230829220, 7.19768271770094931055091007013, 7.51039557222115745086939640217

Graph of the ZZ-function along the critical line