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Properties

Label 8-12e8-1.1-c2e4-0-0
Degree 88
Conductor 429981696429981696
Sign 11
Analytic cond. 237.022237.022
Root an. cond. 1.980831.98083
Motivic weight 22
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  − 18·5-s − 2·7-s + 6·9-s − 18·11-s − 10·13-s + 40·19-s − 18·23-s + 139·25-s + 18·29-s − 38·31-s + 36·35-s + 128·37-s − 126·41-s + 46·43-s − 108·45-s − 54·47-s + 45·49-s + 324·55-s − 126·59-s + 62·61-s − 12·63-s + 180·65-s + 106·67-s − 208·73-s + 36·77-s − 14·79-s − 45·81-s + ⋯
L(s)  = 1  − 3.59·5-s − 2/7·7-s + 2/3·9-s − 1.63·11-s − 0.769·13-s + 2.10·19-s − 0.782·23-s + 5.55·25-s + 0.620·29-s − 1.22·31-s + 1.02·35-s + 3.45·37-s − 3.07·41-s + 1.06·43-s − 2.39·45-s − 1.14·47-s + 0.918·49-s + 5.89·55-s − 2.13·59-s + 1.01·61-s − 0.190·63-s + 2.76·65-s + 1.58·67-s − 2.84·73-s + 0.467·77-s − 0.177·79-s − 5/9·81-s + ⋯

Functional equation

Λ(s)=((216⋅38)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((216⋅38)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅382^{16} \cdot 3^{8}
Sign: 11
Analytic conductor: 237.022237.022
Root analytic conductor: 1.980831.98083
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅38, ( :1,1,1,1), 1)(8,\ 2^{16} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.24385402050.2438540205
L(12)L(\frac12) ≈\approx 0.24385402050.2438540205
L(2)L(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
3C22C_2^2 1−2pT2+p4T4 1 - 2 p T^{2} + p^{4} T^{4}
good5C22C_2^2 (1+9T+52T2+9p2T3+p4T4)2 ( 1 + 9 T + 52 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} )^{2}
7D4×C2D_4\times C_2 1+2T−41T2−106T3−572T4−106p2T5−41p4T6+2p6T7+p8T8 1 + 2 T - 41 T^{2} - 106 T^{3} - 572 T^{4} - 106 p^{2} T^{5} - 41 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8}
11D4×C2D_4\times C_2 1+18T+359T2+4518T3+61428T4+4518p2T5+359p4T6+18p6T7+p8T8 1 + 18 T + 359 T^{2} + 4518 T^{3} + 61428 T^{4} + 4518 p^{2} T^{5} + 359 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8}
13D4×C2D_4\times C_2 1+10T−47T2−1910T3−23852T4−1910p2T5−47p4T6+10p6T7+p8T8 1 + 10 T - 47 T^{2} - 1910 T^{3} - 23852 T^{4} - 1910 p^{2} T^{5} - 47 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8}
17D4×C2D_4\times C_2 1−796T2+294342T4−796p4T6+p8T8 1 - 796 T^{2} + 294342 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8}
19D4D_{4} (1−20T+606T2−20p2T3+p4T4)2 ( 1 - 20 T + 606 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2}
23D4×C2D_4\times C_2 1+18T+1175T2+19206T3+915780T4+19206p2T5+1175p4T6+18p6T7+p8T8 1 + 18 T + 1175 T^{2} + 19206 T^{3} + 915780 T^{4} + 19206 p^{2} T^{5} + 1175 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8}
29D4×C2D_4\times C_2 1−18T+1745T2−29466T3+2063316T4−29466p2T5+1745p4T6−18p6T7+p8T8 1 - 18 T + 1745 T^{2} - 29466 T^{3} + 2063316 T^{4} - 29466 p^{2} T^{5} + 1745 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8}
31D4×C2D_4\times C_2 1+38T−353T2−4750T3+918004T4−4750p2T5−353p4T6+38p6T7+p8T8 1 + 38 T - 353 T^{2} - 4750 T^{3} + 918004 T^{4} - 4750 p^{2} T^{5} - 353 p^{4} T^{6} + 38 p^{6} T^{7} + p^{8} T^{8}
37D4D_{4} (1−64T+3546T2−64p2T3+p4T4)2 ( 1 - 64 T + 3546 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2}
41D4×C2D_4\times C_2 1+126T+9329T2+508662T3+22367460T4+508662p2T5+9329p4T6+126p6T7+p8T8 1 + 126 T + 9329 T^{2} + 508662 T^{3} + 22367460 T^{4} + 508662 p^{2} T^{5} + 9329 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8}
43D4×C2D_4\times C_2 1−46T−1625T2−46pT3+3604p2T4−46p3T5−1625p4T6−46p6T7+p8T8 1 - 46 T - 1625 T^{2} - 46 p T^{3} + 3604 p^{2} T^{4} - 46 p^{3} T^{5} - 1625 p^{4} T^{6} - 46 p^{6} T^{7} + p^{8} T^{8}
47D4×C2D_4\times C_2 1+54T+4751T2+204066T3+11548308T4+204066p2T5+4751p4T6+54p6T7+p8T8 1 + 54 T + 4751 T^{2} + 204066 T^{3} + 11548308 T^{4} + 204066 p^{2} T^{5} + 4751 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8}
53D4×C2D_4\times C_2 1−2236T2−2409114T4−2236p4T6+p8T8 1 - 2236 T^{2} - 2409114 T^{4} - 2236 p^{4} T^{6} + p^{8} T^{8}
59D4×C2D_4\times C_2 1+126T+10535T2+660618T3+33793140T4+660618p2T5+10535p4T6+126p6T7+p8T8 1 + 126 T + 10535 T^{2} + 660618 T^{3} + 33793140 T^{4} + 660618 p^{2} T^{5} + 10535 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8}
61D4×C2D_4\times C_2 1−62T−2615T2+60946T3+13569316T4+60946p2T5−2615p4T6−62p6T7+p8T8 1 - 62 T - 2615 T^{2} + 60946 T^{3} + 13569316 T^{4} + 60946 p^{2} T^{5} - 2615 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8}
67D4×C2D_4\times C_2 1−106T−65T2−246238T3+57123076T4−246238p2T5−65p4T6−106p6T7+p8T8 1 - 106 T - 65 T^{2} - 246238 T^{3} + 57123076 T^{4} - 246238 p^{2} T^{5} - 65 p^{4} T^{6} - 106 p^{6} T^{7} + p^{8} T^{8}
71D4×C2D_4\times C_2 1−12460T2+77194662T4−12460p4T6+p8T8 1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1+104T+11418T2+104p2T3+p4T4)2 ( 1 + 104 T + 11418 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2}
79D4×C2D_4\times C_2 1+14T−10985T2−18214T3+84841444T4−18214p2T5−10985p4T6+14p6T7+p8T8 1 + 14 T - 10985 T^{2} - 18214 T^{3} + 84841444 T^{4} - 18214 p^{2} T^{5} - 10985 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8}
83D4×C2D_4\times C_2 1−378T+72863T2−9538830T3+917456196T4−9538830p2T5+72863p4T6−378p6T7+p8T8 1 - 378 T + 72863 T^{2} - 9538830 T^{3} + 917456196 T^{4} - 9538830 p^{2} T^{5} + 72863 p^{4} T^{6} - 378 p^{6} T^{7} + p^{8} T^{8}
89D4×C2D_4\times C_2 1−8860T2+51019782T4−8860p4T6+p8T8 1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8}
97D4×C2D_4\times C_2 1−14T−8087T2+147490T3−21765356T4+147490p2T5−8087p4T6−14p6T7+p8T8 1 - 14 T - 8087 T^{2} + 147490 T^{3} - 21765356 T^{4} + 147490 p^{2} T^{5} - 8087 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} 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

−9.360792969584319585403220389594, −9.088942613404189964053084332492, −8.851959238502752369320578238840, −8.086147724185144064846346225606, −8.031322823126616782190067622453, −7.983251700254422522710477324063, −7.83479549471841972144866279190, −7.57719767346730460336213020282, −7.28474345165809796533211691652, −6.89168188208284409090052918462, −6.83878757895171972935704247588, −6.24914464636957940769422695232, −5.75716243159949674768872300727, −5.35216484770036810621889406606, −5.22030790239369887826294440978, −4.56958643147551926490471684925, −4.45162547544470042864514415365, −4.24640654739287124804145749730, −3.70185427456299874378854740897, −3.32261068894178789500529427228, −3.23886559272518401240426636071, −2.69396914812535854176709091210, −2.02762009226977346469245053078, −0.925777670434345044508533533931, −0.23865681734576980419040911121, 0.23865681734576980419040911121, 0.925777670434345044508533533931, 2.02762009226977346469245053078, 2.69396914812535854176709091210, 3.23886559272518401240426636071, 3.32261068894178789500529427228, 3.70185427456299874378854740897, 4.24640654739287124804145749730, 4.45162547544470042864514415365, 4.56958643147551926490471684925, 5.22030790239369887826294440978, 5.35216484770036810621889406606, 5.75716243159949674768872300727, 6.24914464636957940769422695232, 6.83878757895171972935704247588, 6.89168188208284409090052918462, 7.28474345165809796533211691652, 7.57719767346730460336213020282, 7.83479549471841972144866279190, 7.983251700254422522710477324063, 8.031322823126616782190067622453, 8.086147724185144064846346225606, 8.851959238502752369320578238840, 9.088942613404189964053084332492, 9.360792969584319585403220389594

Graph of the ZZ-function along the critical line