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Properties

Label 8-42e4-1.1-c4e4-0-0
Degree 88
Conductor 31116963111696
Sign 11
Analytic cond. 355.283355.283
Root an. cond. 2.083632.08363
Motivic weight 44
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  − 18·3-s − 8·4-s − 66·5-s − 70·7-s + 189·9-s − 162·11-s + 144·12-s + 1.18e3·15-s − 204·17-s − 444·19-s + 528·20-s + 1.26e3·21-s + 312·23-s + 1.31e3·25-s − 1.45e3·27-s + 560·28-s + 2.72e3·29-s − 3.78e3·31-s + 2.91e3·33-s + 4.62e3·35-s − 1.51e3·36-s + 1.39e3·37-s − 632·43-s + 1.29e3·44-s − 1.24e4·45-s − 7.89e3·47-s + 2.40e3·49-s + ⋯
L(s)  = 1  − 2·3-s − 1/2·4-s − 2.63·5-s − 1.42·7-s + 7/3·9-s − 1.33·11-s + 12-s + 5.27·15-s − 0.705·17-s − 1.22·19-s + 1.31·20-s + 20/7·21-s + 0.589·23-s + 2.10·25-s − 2·27-s + 5/7·28-s + 3.23·29-s − 3.93·31-s + 2.67·33-s + 3.77·35-s − 7/6·36-s + 1.01·37-s − 0.341·43-s + 0.669·44-s − 6.15·45-s − 3.57·47-s + 49-s + ⋯

Functional equation

Λ(s)=(3111696s/2 ΓC(s)4 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(3111696s/2 ΓC(s+2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 31116963111696    =    24⋅34⋅742^{4} \cdot 3^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 355.283355.283
Root analytic conductor: 2.083632.08363
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 3111696, ( :2,2,2,2), 1)(8,\ 3111696,\ (\ :2, 2, 2, 2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.11424544760.1142454476
L(12)L(\frac12) ≈\approx 0.11424544760.1142454476
L(3)L(3) 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)
bad2C22C_2^2 1+p3T2+p6T4 1 + p^{3} T^{2} + p^{6} T^{4}
3C2C_2 (1+p2T+p3T2)2 ( 1 + p^{2} T + p^{3} T^{2} )^{2}
7C22C_2^2 1+10pT+51p2T2+10p5T3+p8T4 1 + 10 p T + 51 p^{2} T^{2} + 10 p^{5} T^{3} + p^{8} T^{4}
good5D4×C2D_4\times C_2 1+66T+3041T2+104874T3+3041796T4+104874p4T5+3041p8T6+66p12T7+p16T8 1 + 66 T + 3041 T^{2} + 104874 T^{3} + 3041796 T^{4} + 104874 p^{4} T^{5} + 3041 p^{8} T^{6} + 66 p^{12} T^{7} + p^{16} T^{8}
11D4×C2D_4\times C_2 1+162T−9311T2+1016226T3+665560740T4+1016226p4T5−9311p8T6+162p12T7+p16T8 1 + 162 T - 9311 T^{2} + 1016226 T^{3} + 665560740 T^{4} + 1016226 p^{4} T^{5} - 9311 p^{8} T^{6} + 162 p^{12} T^{7} + p^{16} T^{8}
13D4×C2D_4\times C_2 1−50980T2+1849726854T4−50980p8T6+p16T8 1 - 50980 T^{2} + 1849726854 T^{4} - 50980 p^{8} T^{6} + p^{16} T^{8}
17D4×C2D_4\times C_2 1+12pT+542p2T2+5928p3T3+174387p4T4+5928p7T5+542p10T6+12p13T7+p16T8 1 + 12 p T + 542 p^{2} T^{2} + 5928 p^{3} T^{3} + 174387 p^{4} T^{4} + 5928 p^{7} T^{5} + 542 p^{10} T^{6} + 12 p^{13} T^{7} + p^{16} T^{8}
19D4×C2D_4\times C_2 1+444T+315038T2+110700744T3+53743544787T4+110700744p4T5+315038p8T6+444p12T7+p16T8 1 + 444 T + 315038 T^{2} + 110700744 T^{3} + 53743544787 T^{4} + 110700744 p^{4} T^{5} + 315038 p^{8} T^{6} + 444 p^{12} T^{7} + p^{16} T^{8}
23D4×C2D_4\times C_2 1−312T+262414T2+226122624T3−78303722685T4+226122624p4T5+262414p8T6−312p12T7+p16T8 1 - 312 T + 262414 T^{2} + 226122624 T^{3} - 78303722685 T^{4} + 226122624 p^{4} T^{5} + 262414 p^{8} T^{6} - 312 p^{12} T^{7} + p^{16} T^{8}
29D4D_{4} (1−1362T+1874795T2−1362p4T3+p8T4)2 ( 1 - 1362 T + 1874795 T^{2} - 1362 p^{4} T^{3} + p^{8} T^{4} )^{2}
31D4×C2D_4\times C_2 1+3786T+7637801T2+10827464034T3+11738480198292T4+10827464034p4T5+7637801p8T6+3786p12T7+p16T8 1 + 3786 T + 7637801 T^{2} + 10827464034 T^{3} + 11738480198292 T^{4} + 10827464034 p^{4} T^{5} + 7637801 p^{8} T^{6} + 3786 p^{12} T^{7} + p^{16} T^{8}
37D4×C2D_4\times C_2 1−1396T−2268278T2−654405712T3+10619007407539T4−654405712p4T5−2268278p8T6−1396p12T7+p16T8 1 - 1396 T - 2268278 T^{2} - 654405712 T^{3} + 10619007407539 T^{4} - 654405712 p^{4} T^{5} - 2268278 p^{8} T^{6} - 1396 p^{12} T^{7} + p^{16} T^{8}
41D4×C2D_4\times C_2 1−2317228T2+5437495988070T4−2317228p8T6+p16T8 1 - 2317228 T^{2} + 5437495988070 T^{4} - 2317228 p^{8} T^{6} + p^{16} T^{8}
43D4D_{4} (1+316T−3534234T2+316p4T3+p8T4)2 ( 1 + 316 T - 3534234 T^{2} + 316 p^{4} T^{3} + p^{8} T^{4} )^{2}
47D4×C2D_4\times C_2 1+168pT+32962802T2+2046329040pT3+225964882234371T4+2046329040p5T5+32962802p8T6+168p13T7+p16T8 1 + 168 p T + 32962802 T^{2} + 2046329040 p T^{3} + 225964882234371 T^{4} + 2046329040 p^{5} T^{5} + 32962802 p^{8} T^{6} + 168 p^{13} T^{7} + p^{16} T^{8}
53D4×C2D_4\times C_2 1+1038T−12178631T2−2620832706T3+104962282583700T4−2620832706p4T5−12178631p8T6+1038p12T7+p16T8 1 + 1038 T - 12178631 T^{2} - 2620832706 T^{3} + 104962282583700 T^{4} - 2620832706 p^{4} T^{5} - 12178631 p^{8} T^{6} + 1038 p^{12} T^{7} + p^{16} T^{8}
59D4×C2D_4\times C_2 1+966T+14295473T2+13508950686T3+52502722474692T4+13508950686p4T5+14295473p8T6+966p12T7+p16T8 1 + 966 T + 14295473 T^{2} + 13508950686 T^{3} + 52502722474692 T^{4} + 13508950686 p^{4} T^{5} + 14295473 p^{8} T^{6} + 966 p^{12} T^{7} + p^{16} T^{8}
61D4×C2D_4\times C_2 1−5088T+37832738T2−148587357120T3+780615710940387T4−148587357120p4T5+37832738p8T6−5088p12T7+p16T8 1 - 5088 T + 37832738 T^{2} - 148587357120 T^{3} + 780615710940387 T^{4} - 148587357120 p^{4} T^{5} + 37832738 p^{8} T^{6} - 5088 p^{12} T^{7} + p^{16} T^{8}
67D4×C2D_4\times C_2 1−14600T+121457326T2−750446307200T3+3707899788833635T4−750446307200p4T5+121457326p8T6−14600p12T7+p16T8 1 - 14600 T + 121457326 T^{2} - 750446307200 T^{3} + 3707899788833635 T^{4} - 750446307200 p^{4} T^{5} + 121457326 p^{8} T^{6} - 14600 p^{12} T^{7} + p^{16} T^{8}
71D4D_{4} (1+4848T+52412546T2+4848p4T3+p8T4)2 ( 1 + 4848 T + 52412546 T^{2} + 4848 p^{4} T^{3} + p^{8} T^{4} )^{2}
73D4×C2D_4\times C_2 1−22584T+263311922T2−2107077488880T3+12726401415363651T4−2107077488880p4T5+263311922p8T6−22584p12T7+p16T8 1 - 22584 T + 263311922 T^{2} - 2107077488880 T^{3} + 12726401415363651 T^{4} - 2107077488880 p^{4} T^{5} + 263311922 p^{8} T^{6} - 22584 p^{12} T^{7} + p^{16} T^{8}
79D4×C2D_4\times C_2 1−3974T+5512993T2+268723783546T3−2026562936732828T4+268723783546p4T5+5512993p8T6−3974p12T7+p16T8 1 - 3974 T + 5512993 T^{2} + 268723783546 T^{3} - 2026562936732828 T^{4} + 268723783546 p^{4} T^{5} + 5512993 p^{8} T^{6} - 3974 p^{12} T^{7} + p^{16} T^{8}
83D4×C2D_4\times C_2 1−1452506pT2+7047387074675283T4−1452506p9T6+p16T8 1 - 1452506 p T^{2} + 7047387074675283 T^{4} - 1452506 p^{9} T^{6} + p^{16} T^{8}
89D4×C2D_4\times C_2 1+33156T+547930046T2+6017480251704T3+51993281156793267T4+6017480251704p4T5+547930046p8T6+33156p12T7+p16T8 1 + 33156 T + 547930046 T^{2} + 6017480251704 T^{3} + 51993281156793267 T^{4} + 6017480251704 p^{4} T^{5} + 547930046 p^{8} T^{6} + 33156 p^{12} T^{7} + p^{16} T^{8}
97D4×C2D_4\times C_2 1−347883310T2+45930093674217747T4−347883310p8T6+p16T8 1 - 347883310 T^{2} + 45930093674217747 T^{4} - 347883310 p^{8} T^{6} + p^{16} 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

−11.21113520241450686464224636099, −11.07583073519812671282175951655, −10.99754582046008135712481247391, −10.22762298009413030304431728961, −10.05579358654567040896560908666, −9.701404869765280392685935704482, −9.443278270233122208696910349472, −8.616198391398499759997982866571, −8.513461754460515562636713109758, −8.095716347713537995701107822048, −7.81954727127925057209467155950, −7.36786516011411637452826745338, −6.84024858086050808473345619818, −6.62621496979631768334268126376, −6.51380775269341677071313114921, −5.72389449553646269190787369851, −5.36751440368906582812486783957, −4.88653222506638658314314640790, −4.51235265272485890990695384286, −4.15999634413400173504192926364, −3.43155639102114000304224415141, −3.39742501952175080326639774109, −2.18237728783842648973651337013, −0.44197560749307898514498397904, −0.35861722902111823352446220949, 0.35861722902111823352446220949, 0.44197560749307898514498397904, 2.18237728783842648973651337013, 3.39742501952175080326639774109, 3.43155639102114000304224415141, 4.15999634413400173504192926364, 4.51235265272485890990695384286, 4.88653222506638658314314640790, 5.36751440368906582812486783957, 5.72389449553646269190787369851, 6.51380775269341677071313114921, 6.62621496979631768334268126376, 6.84024858086050808473345619818, 7.36786516011411637452826745338, 7.81954727127925057209467155950, 8.095716347713537995701107822048, 8.513461754460515562636713109758, 8.616198391398499759997982866571, 9.443278270233122208696910349472, 9.701404869765280392685935704482, 10.05579358654567040896560908666, 10.22762298009413030304431728961, 10.99754582046008135712481247391, 11.07583073519812671282175951655, 11.21113520241450686464224636099

Graph of the ZZ-function along the critical line