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Properties

Label 8-24e8-1.1-c1e4-0-8
Degree 88
Conductor 110075314176110075314176
Sign 11
Analytic cond. 447.505447.505
Root an. cond. 2.144612.14461
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 8·5-s + 6·7-s + 21·9-s + 10·11-s − 10·13-s − 48·15-s − 16·17-s + 6·19-s + 36·21-s − 6·23-s + 44·25-s + 54·27-s − 6·29-s + 8·31-s + 60·33-s − 48·35-s + 12·37-s − 60·39-s + 16·43-s − 168·45-s + 2·47-s + 8·49-s − 96·51-s − 16·53-s − 80·55-s + 36·57-s + ⋯
L(s)  = 1  + 3.46·3-s − 3.57·5-s + 2.26·7-s + 7·9-s + 3.01·11-s − 2.77·13-s − 12.3·15-s − 3.88·17-s + 1.37·19-s + 7.85·21-s − 1.25·23-s + 44/5·25-s + 10.3·27-s − 1.11·29-s + 1.43·31-s + 10.4·33-s − 8.11·35-s + 1.97·37-s − 9.60·39-s + 2.43·43-s − 25.0·45-s + 0.291·47-s + 8/7·49-s − 13.4·51-s − 2.19·53-s − 10.7·55-s + 4.76·57-s + ⋯

Functional equation

Λ(s)=((224⋅38)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅38)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅382^{24} \cdot 3^{8}
Sign: 11
Analytic conductor: 447.505447.505
Root analytic conductor: 2.144612.14461
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅38, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 6.7330233206.733023320
L(12)L(\frac12) ≈\approx 6.7330233206.733023320
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−pT+pT2)2 ( 1 - p T + p T^{2} )^{2}
good5D4×C2D_4\times C_2 1+8T+4pT2−4T3−89T4−4pT5+4p3T6+8p3T7+p4T8 1 + 8 T + 4 p T^{2} - 4 T^{3} - 89 T^{4} - 4 p T^{5} + 4 p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
7D4×C2D_4\times C_2 1−6T+4pT2−96T3+291T4−96pT5+4p3T6−6p3T7+p4T8 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1−10T+41T2−82T3+136T4−82pT5+41p2T6−10p3T7+p4T8 1 - 10 T + 41 T^{2} - 82 T^{3} + 136 T^{4} - 82 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
13C2C_2×\timesC22C_2^2 (1+5T+pT2)2(1+23T2+p2T4) ( 1 + 5 T + p T^{2} )^{2}( 1 + 23 T^{2} + p^{2} T^{4} )
17C22C_2^2 (1+8T+47T2+8pT3+p2T4)2 ( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
19D4×C2D_4\times C_2 1−6T+18T2−132T3+959T4−132pT5+18p2T6−6p3T7+p4T8 1 - 6 T + 18 T^{2} - 132 T^{3} + 959 T^{4} - 132 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1+6T+52T2+240T3+1347T4+240pT5+52p2T6+6p3T7+p4T8 1 + 6 T + 52 T^{2} + 240 T^{3} + 1347 T^{4} + 240 p T^{5} + 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
29C23C_2^3 1+6T+18T2+36T3−457T4+36pT5+18p2T6+6p3T7+p4T8 1 + 6 T + 18 T^{2} + 36 T^{3} - 457 T^{4} + 36 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
31D4×C2D_4\times C_2 1−8T−2T2−32T3+1411T4−32pT5−2p2T6−8p3T7+p4T8 1 - 8 T - 2 T^{2} - 32 T^{3} + 1411 T^{4} - 32 p T^{5} - 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
37D4×C2D_4\times C_2 1−12T+72T2−588T3+4658T4−588pT5+72p2T6−12p3T7+p4T8 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 588 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
41C23C_2^3 1+73T2+3648T4+73p2T6+p4T8 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8}
43D4×C2D_4\times C_2 1−16T+65T2+624T3−8092T4+624pT5+65p2T6−16p3T7+p4T8 1 - 16 T + 65 T^{2} + 624 T^{3} - 8092 T^{4} + 624 p T^{5} + 65 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−2T−16T2+148T3−1997T4+148pT5−16p2T6−2p3T7+p4T8 1 - 2 T - 16 T^{2} + 148 T^{3} - 1997 T^{4} + 148 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
53D4×C2D_4\times C_2 1+16T+128T2+976T3+7378T4+976pT5+128p2T6+16p3T7+p4T8 1 + 16 T + 128 T^{2} + 976 T^{3} + 7378 T^{4} + 976 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
59D4×C2D_4\times C_2 1+6T+45T2−594T3−3376T4−594pT5+45p2T6+6p3T7+p4T8 1 + 6 T + 45 T^{2} - 594 T^{3} - 3376 T^{4} - 594 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1+12T+180T2+1596T3+15143T4+1596pT5+180p2T6+12p3T7+p4T8 1 + 12 T + 180 T^{2} + 1596 T^{3} + 15143 T^{4} + 1596 p T^{5} + 180 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1+16T+113T2+384T3−172T4+384pT5+113p2T6+16p3T7+p4T8 1 + 16 T + 113 T^{2} + 384 T^{3} - 172 T^{4} + 384 p T^{5} + 113 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
71D4×C2D_4\times C_2 1−156T2+13094T4−156p2T6+p4T8 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8}
73D4×C2D_4\times C_2 1−158T2+16131T4−158p2T6+p4T8 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8}
79C22C_2^2 (1−12T+65T2−12pT3+p2T4)2 ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
83C23C_2^3 1−2T+2T2+328T3−7217T4+328pT5+2p2T6−2p3T7+p4T8 1 - 2 T + 2 T^{2} + 328 T^{3} - 7217 T^{4} + 328 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
89C22C_2^2 (1−174T2+p2T4)2 ( 1 - 174 T^{2} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1+20T+109T2+20pT3+376pT4+20p2T5+109p2T6+20p3T7+p4T8 1 + 20 T + 109 T^{2} + 20 p T^{3} + 376 p T^{4} + 20 p^{2} T^{5} + 109 p^{2} T^{6} + 20 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

−7.87717361387222809149298613794, −7.47500259637307731378188261310, −7.45522003395365809876340972527, −7.26120610036209112453457874834, −7.20392190230465880518958012204, −6.91859964322026948445517198573, −6.44067288560019112536254488185, −6.26041863570242261577086910155, −6.10517468183462305665122457970, −5.04513515440582730632390454204, −4.80262701270519440538109749946, −4.58169251470589773119412884980, −4.52894018041374925835526999335, −4.41212321747056481783418730999, −4.22426791487876113833159390421, −4.07698259689205053550646436199, −3.49327055423267326069165901655, −3.46159142506378581515616693618, −3.01140495584972192352855143522, −2.79285429412326055230635552274, −2.40603293811004784736397263174, −1.93184635932870062166391627912, −1.91493704273533030447488103580, −1.26848366388194224536007203471, −0.64718781797248432170085297969, 0.64718781797248432170085297969, 1.26848366388194224536007203471, 1.91493704273533030447488103580, 1.93184635932870062166391627912, 2.40603293811004784736397263174, 2.79285429412326055230635552274, 3.01140495584972192352855143522, 3.46159142506378581515616693618, 3.49327055423267326069165901655, 4.07698259689205053550646436199, 4.22426791487876113833159390421, 4.41212321747056481783418730999, 4.52894018041374925835526999335, 4.58169251470589773119412884980, 4.80262701270519440538109749946, 5.04513515440582730632390454204, 6.10517468183462305665122457970, 6.26041863570242261577086910155, 6.44067288560019112536254488185, 6.91859964322026948445517198573, 7.20392190230465880518958012204, 7.26120610036209112453457874834, 7.45522003395365809876340972527, 7.47500259637307731378188261310, 7.87717361387222809149298613794

Graph of the ZZ-function along the critical line