Properties

Label 8-13e4-1.1-c9e4-0-0
Degree 88
Conductor 2856128561
Sign 11
Analytic cond. 2009.662009.66
Root an. cond. 2.587552.58755
Motivic weight 99
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 33·2-s − 163·3-s + 235·4-s + 471·5-s + 5.37e3·6-s − 1.12e4·7-s − 5.32e3·8-s − 4.10e4·9-s − 1.55e4·10-s − 4.01e4·11-s − 3.83e4·12-s − 1.14e5·13-s + 3.70e5·14-s − 7.67e4·15-s + 2.42e5·16-s + 7.87e4·17-s + 1.35e6·18-s + 2.09e5·19-s + 1.10e5·20-s + 1.83e6·21-s + 1.32e6·22-s − 4.25e6·23-s + 8.67e5·24-s − 5.24e6·25-s + 3.77e6·26-s + 8.51e6·27-s − 2.64e6·28-s + ⋯
L(s)  = 1  − 1.45·2-s − 1.16·3-s + 0.458·4-s + 0.337·5-s + 1.69·6-s − 1.76·7-s − 0.459·8-s − 2.08·9-s − 0.491·10-s − 0.826·11-s − 0.533·12-s − 1.10·13-s + 2.58·14-s − 0.391·15-s + 0.925·16-s + 0.228·17-s + 3.04·18-s + 0.369·19-s + 0.154·20-s + 2.05·21-s + 1.20·22-s − 3.17·23-s + 0.534·24-s − 2.68·25-s + 1.61·26-s + 3.08·27-s − 0.812·28-s + ⋯

Functional equation

Λ(s)=(28561s/2ΓC(s)4L(s)=(Λ(10s)\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(28561s/2ΓC(s+9/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 2856128561    =    13413^{4}
Sign: 11
Analytic conductor: 2009.662009.66
Root analytic conductor: 2.587552.58755
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 28561, ( :9/2,9/2,9/2,9/2), 1)(8,\ 28561,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad13C1C_1 (1+p4T)4 ( 1 + p^{4} T )^{4}
good2C2S4C_2 \wr S_4 1+33T+427pT2+3219p3T3+9097p6T4+3219p12T5+427p19T6+33p27T7+p36T8 1 + 33 T + 427 p T^{2} + 3219 p^{3} T^{3} + 9097 p^{6} T^{4} + 3219 p^{12} T^{5} + 427 p^{19} T^{6} + 33 p^{27} T^{7} + p^{36} T^{8}
3C2S4C_2 \wr S_4 1+163T+67627T2+3067592pT3+212179120p2T4+3067592p10T5+67627p18T6+163p27T7+p36T8 1 + 163 T + 67627 T^{2} + 3067592 p T^{3} + 212179120 p^{2} T^{4} + 3067592 p^{10} T^{5} + 67627 p^{18} T^{6} + 163 p^{27} T^{7} + p^{36} T^{8}
5C2S4C_2 \wr S_4 1471T+5467249T22978826294T3+2762975487438pT42978826294p9T5+5467249p18T6471p27T7+p36T8 1 - 471 T + 5467249 T^{2} - 2978826294 T^{3} + 2762975487438 p T^{4} - 2978826294 p^{9} T^{5} + 5467249 p^{18} T^{6} - 471 p^{27} T^{7} + p^{36} T^{8}
7C2S4C_2 \wr S_4 1+11241T+18891399pT2+2792105460p3T3+22711650939506p3T4+2792105460p12T5+18891399p19T6+11241p27T7+p36T8 1 + 11241 T + 18891399 p T^{2} + 2792105460 p^{3} T^{3} + 22711650939506 p^{3} T^{4} + 2792105460 p^{12} T^{5} + 18891399 p^{19} T^{6} + 11241 p^{27} T^{7} + p^{36} T^{8}
11C2S4C_2 \wr S_4 1+40140T+5207212188T2+260939722519836T3+16186230268309124390T4+260939722519836p9T5+5207212188p18T6+40140p27T7+p36T8 1 + 40140 T + 5207212188 T^{2} + 260939722519836 T^{3} + 16186230268309124390 T^{4} + 260939722519836 p^{9} T^{5} + 5207212188 p^{18} T^{6} + 40140 p^{27} T^{7} + p^{36} T^{8}
17C2S4C_2 \wr S_4 178717T+179543541353T2+34209748146126354T3+ 1 - 78717 T + 179543541353 T^{2} + 34209748146126354 T^{3} + 16 ⁣ ⁣8216\!\cdots\!82T4+34209748146126354p9T5+179543541353p18T678717p27T7+p36T8 T^{4} + 34209748146126354 p^{9} T^{5} + 179543541353 p^{18} T^{6} - 78717 p^{27} T^{7} + p^{36} T^{8}
19C2S4C_2 \wr S_4 1209664T+58556387148pT29405441041308416pT3+ 1 - 209664 T + 58556387148 p T^{2} - 9405441041308416 p T^{3} + 51 ⁣ ⁣2251\!\cdots\!22T49405441041308416p10T5+58556387148p19T6209664p27T7+p36T8 T^{4} - 9405441041308416 p^{10} T^{5} + 58556387148 p^{19} T^{6} - 209664 p^{27} T^{7} + p^{36} T^{8}
23C2S4C_2 \wr S_4 1+4257444T+7366209360108T2+5287090655000044788T3+ 1 + 4257444 T + 7366209360108 T^{2} + 5287090655000044788 T^{3} + 22 ⁣ ⁣3422\!\cdots\!34T4+5287090655000044788p9T5+7366209360108p18T6+4257444p27T7+p36T8 T^{4} + 5287090655000044788 p^{9} T^{5} + 7366209360108 p^{18} T^{6} + 4257444 p^{27} T^{7} + p^{36} T^{8}
29C2S4C_2 \wr S_4 1+1647936T+37315906740828T2+63599271454922849280T3+ 1 + 1647936 T + 37315906740828 T^{2} + 63599271454922849280 T^{3} + 66 ⁣ ⁣7466\!\cdots\!74T4+63599271454922849280p9T5+37315906740828p18T6+1647936p27T7+p36T8 T^{4} + 63599271454922849280 p^{9} T^{5} + 37315906740828 p^{18} T^{6} + 1647936 p^{27} T^{7} + p^{36} T^{8}
31C2S4C_2 \wr S_4 1+11366002T+118630502542644T2+ 1 + 11366002 T + 118630502542644 T^{2} + 68 ⁣ ⁣1868\!\cdots\!18T3+ T^{3} + 42 ⁣ ⁣0642\!\cdots\!06T4+ T^{4} + 68 ⁣ ⁣1868\!\cdots\!18p9T5+118630502542644p18T6+11366002p27T7+p36T8 p^{9} T^{5} + 118630502542644 p^{18} T^{6} + 11366002 p^{27} T^{7} + p^{36} T^{8}
37C2S4C_2 \wr S_4 14636891T+361056879801945T2 1 - 4636891 T + 361056879801945 T^{2} - 14 ⁣ ⁣9414\!\cdots\!94T3+ T^{3} + 66 ⁣ ⁣3466\!\cdots\!34T4 T^{4} - 14 ⁣ ⁣9414\!\cdots\!94p9T5+361056879801945p18T64636891p27T7+p36T8 p^{9} T^{5} + 361056879801945 p^{18} T^{6} - 4636891 p^{27} T^{7} + p^{36} T^{8}
41C2S4C_2 \wr S_4 113859538T+1246904510238356T2 1 - 13859538 T + 1246904510238356 T^{2} - 13 ⁣ ⁣8613\!\cdots\!86T3+ T^{3} + 60 ⁣ ⁣4260\!\cdots\!42T4 T^{4} - 13 ⁣ ⁣8613\!\cdots\!86p9T5+1246904510238356p18T613859538p27T7+p36T8 p^{9} T^{5} + 1246904510238356 p^{18} T^{6} - 13859538 p^{27} T^{7} + p^{36} T^{8}
43C2S4C_2 \wr S_4 1+33368081T+1484927456162487T2+ 1 + 33368081 T + 1484927456162487 T^{2} + 42 ⁣ ⁣1242\!\cdots\!12T3+ T^{3} + 10 ⁣ ⁣2410\!\cdots\!24T4+ T^{4} + 42 ⁣ ⁣1242\!\cdots\!12p9T5+1484927456162487p18T6+33368081p27T7+p36T8 p^{9} T^{5} + 1484927456162487 p^{18} T^{6} + 33368081 p^{27} T^{7} + p^{36} T^{8}
47C2S4C_2 \wr S_4 1+3943005T+2343750668926273T2+ 1 + 3943005 T + 2343750668926273 T^{2} + 81 ⁣ ⁣9681\!\cdots\!96T3+ T^{3} + 37 ⁣ ⁣5437\!\cdots\!54T4+ T^{4} + 81 ⁣ ⁣9681\!\cdots\!96p9T5+2343750668926273p18T6+3943005p27T7+p36T8 p^{9} T^{5} + 2343750668926273 p^{18} T^{6} + 3943005 p^{27} T^{7} + p^{36} T^{8}
53C2S4C_2 \wr S_4 1+171019326T+20398534023554412T2+ 1 + 171019326 T + 20398534023554412 T^{2} + 16 ⁣ ⁣8616\!\cdots\!86T3+ T^{3} + 11 ⁣ ⁣6611\!\cdots\!66T4+ T^{4} + 16 ⁣ ⁣8616\!\cdots\!86p9T5+20398534023554412p18T6+171019326p27T7+p36T8 p^{9} T^{5} + 20398534023554412 p^{18} T^{6} + 171019326 p^{27} T^{7} + p^{36} T^{8}
59C2S4C_2 \wr S_4 1+63389388T+18604706307570428T2+ 1 + 63389388 T + 18604706307570428 T^{2} + 24 ⁣ ⁣6824\!\cdots\!68T3+ T^{3} + 14 ⁣ ⁣7414\!\cdots\!74T4+ T^{4} + 24 ⁣ ⁣6824\!\cdots\!68p9T5+18604706307570428p18T6+63389388p27T7+p36T8 p^{9} T^{5} + 18604706307570428 p^{18} T^{6} + 63389388 p^{27} T^{7} + p^{36} T^{8}
61C2S4C_2 \wr S_4 177050190T+39401474083082580T2 1 - 77050190 T + 39401474083082580 T^{2} - 26 ⁣ ⁣8626\!\cdots\!86T3+ T^{3} + 65 ⁣ ⁣7065\!\cdots\!70T4 T^{4} - 26 ⁣ ⁣8626\!\cdots\!86p9T5+39401474083082580p18T677050190p27T7+p36T8 p^{9} T^{5} + 39401474083082580 p^{18} T^{6} - 77050190 p^{27} T^{7} + p^{36} T^{8}
67C2S4C_2 \wr S_4 1+41174072T+67202484458082900T2+ 1 + 41174072 T + 67202484458082900 T^{2} + 19 ⁣ ⁣2819\!\cdots\!28T3+ T^{3} + 23 ⁣ ⁣4623\!\cdots\!46T4+ T^{4} + 19 ⁣ ⁣2819\!\cdots\!28p9T5+67202484458082900p18T6+41174072p27T7+p36T8 p^{9} T^{5} + 67202484458082900 p^{18} T^{6} + 41174072 p^{27} T^{7} + p^{36} T^{8}
71C2S4C_2 \wr S_4 1252460989T+144403365822266409T2 1 - 252460989 T + 144403365822266409 T^{2} - 44 ⁣ ⁣6844\!\cdots\!68pT3+ p T^{3} + 90 ⁣ ⁣9090\!\cdots\!90T4 T^{4} - 44 ⁣ ⁣6844\!\cdots\!68p10T5+144403365822266409p18T6252460989p27T7+p36T8 p^{10} T^{5} + 144403365822266409 p^{18} T^{6} - 252460989 p^{27} T^{7} + p^{36} T^{8}
73C2S4C_2 \wr S_4 1594415068T+287796385433349156T2 1 - 594415068 T + 287796385433349156 T^{2} - 92 ⁣ ⁣7692\!\cdots\!76T3+ T^{3} + 26 ⁣ ⁣4626\!\cdots\!46T4 T^{4} - 92 ⁣ ⁣7692\!\cdots\!76p9T5+287796385433349156p18T6594415068p27T7+p36T8 p^{9} T^{5} + 287796385433349156 p^{18} T^{6} - 594415068 p^{27} T^{7} + p^{36} T^{8}
79C2S4C_2 \wr S_4 1115998984T+371265024783771132T2 1 - 115998984 T + 371265024783771132 T^{2} - 33 ⁣ ⁣8033\!\cdots\!80T3+ T^{3} + 62 ⁣ ⁣5862\!\cdots\!58T4 T^{4} - 33 ⁣ ⁣8033\!\cdots\!80p9T5+371265024783771132p18T6115998984p27T7+p36T8 p^{9} T^{5} + 371265024783771132 p^{18} T^{6} - 115998984 p^{27} T^{7} + p^{36} T^{8}
83C2S4C_2 \wr S_4 1+79577862T+6204891761421028pT2+ 1 + 79577862 T + 6204891761421028 p T^{2} + 86 ⁣ ⁣4686\!\cdots\!46T3+ T^{3} + 12 ⁣ ⁣3812\!\cdots\!38T4+ T^{4} + 86 ⁣ ⁣4686\!\cdots\!46p9T5+6204891761421028p19T6+79577862p27T7+p36T8 p^{9} T^{5} + 6204891761421028 p^{19} T^{6} + 79577862 p^{27} T^{7} + p^{36} T^{8}
89C2S4C_2 \wr S_4 1+1152240276T+746897787143209636T2+ 1 + 1152240276 T + 746897787143209636 T^{2} + 27 ⁣ ⁣0027\!\cdots\!00T3 T^{3} - 70 ⁣ ⁣1470\!\cdots\!14T4+ T^{4} + 27 ⁣ ⁣0027\!\cdots\!00p9T5+746897787143209636p18T6+1152240276p27T7+p36T8 p^{9} T^{5} + 746897787143209636 p^{18} T^{6} + 1152240276 p^{27} T^{7} + p^{36} T^{8}
97C2S4C_2 \wr S_4 11049098084T+2374666548574190388T2 1 - 1049098084 T + 2374666548574190388 T^{2} - 23 ⁣ ⁣6023\!\cdots\!60T3+ T^{3} + 24 ⁣ ⁣7424\!\cdots\!74T4 T^{4} - 23 ⁣ ⁣6023\!\cdots\!60p9T5+2374666548574190388p18T61049098084p27T7+p36T8 p^{9} T^{5} + 2374666548574190388 p^{18} T^{6} - 1049098084 p^{27} T^{7} + p^{36} T^{8}
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   L(s)=p j=18(1αj,pps)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

−13.95928460430890317870092855744, −12.97106901514597117453914352925, −12.85445491453793972570809894225, −12.43626443260267289556638154263, −11.83918470565123820424209471048, −11.75124371675164354200421837086, −11.31969655516604555701088514367, −10.94595480247553116440448571098, −10.26495302653789196245989292064, −9.892948414715274431333698709278, −9.608986960805973076664105819727, −9.274698042266345077491294788431, −9.196526681358649258536784772497, −8.044205686931871745573286906803, −8.039987945857310034350189533328, −7.82905030775083749862309007042, −6.57639634467340707158950518251, −6.31081068795648558770452480345, −5.84923773055124293501359703709, −5.54522832179349602028947311014, −5.19631160721890757832532363900, −3.73592644965051715793501526344, −3.38532871414856603989847282361, −2.51114059648194216727569569543, −1.97447730592761554791926188116, 0, 0, 0, 0, 1.97447730592761554791926188116, 2.51114059648194216727569569543, 3.38532871414856603989847282361, 3.73592644965051715793501526344, 5.19631160721890757832532363900, 5.54522832179349602028947311014, 5.84923773055124293501359703709, 6.31081068795648558770452480345, 6.57639634467340707158950518251, 7.82905030775083749862309007042, 8.039987945857310034350189533328, 8.044205686931871745573286906803, 9.196526681358649258536784772497, 9.274698042266345077491294788431, 9.608986960805973076664105819727, 9.892948414715274431333698709278, 10.26495302653789196245989292064, 10.94595480247553116440448571098, 11.31969655516604555701088514367, 11.75124371675164354200421837086, 11.83918470565123820424209471048, 12.43626443260267289556638154263, 12.85445491453793972570809894225, 12.97106901514597117453914352925, 13.95928460430890317870092855744

Graph of the ZZ-function along the critical line