Properties

Label 8-384e4-1.1-c9e4-0-3
Degree 88
Conductor 2174327193621743271936
Sign 11
Analytic cond. 1.52994×1091.52994\times 10^{9}
Root an. cond. 14.063214.0632
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  − 324·3-s − 1.72e3·5-s + 4.84e3·7-s + 6.56e4·9-s − 1.58e4·11-s − 8.24e4·13-s + 5.59e5·15-s + 1.65e5·17-s − 5.39e5·19-s − 1.56e6·21-s + 7.29e5·23-s − 2.30e6·25-s − 1.06e7·27-s − 1.85e6·29-s + 3.19e6·31-s + 5.12e6·33-s − 8.36e6·35-s − 4.18e6·37-s + 2.67e7·39-s + 2.27e5·41-s − 1.60e6·43-s − 1.13e8·45-s + 1.80e7·47-s − 4.01e7·49-s − 5.37e7·51-s − 2.94e7·53-s + 2.73e7·55-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.23·5-s + 0.761·7-s + 10/3·9-s − 0.325·11-s − 0.800·13-s + 2.85·15-s + 0.481·17-s − 0.950·19-s − 1.75·21-s + 0.543·23-s − 1.17·25-s − 3.84·27-s − 0.485·29-s + 0.621·31-s + 0.752·33-s − 0.942·35-s − 0.367·37-s + 1.84·39-s + 0.0125·41-s − 0.0713·43-s − 4.12·45-s + 0.539·47-s − 0.994·49-s − 1.11·51-s − 0.512·53-s + 0.402·55-s + ⋯

Functional equation

Λ(s)=((22834)s/2ΓC(s)4L(s)=(Λ(10s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}
Λ(s)=((22834)s/2ΓC(s+9/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228342^{28} \cdot 3^{4}
Sign: 11
Analytic conductor: 1.52994×1091.52994\times 10^{9}
Root analytic conductor: 14.063214.0632
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 22834, ( :9/2,9/2,9/2,9/2), 1)(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 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)
bad2 1 1
3C1C_1 (1+p4T)4 ( 1 + p^{4} T )^{4}
good5C2S4C_2 \wr S_4 1+1728T+5286836T2+1344360512pT3+585667235526p2T4+1344360512p10T5+5286836p18T6+1728p27T7+p36T8 1 + 1728 T + 5286836 T^{2} + 1344360512 p T^{3} + 585667235526 p^{2} T^{4} + 1344360512 p^{10} T^{5} + 5286836 p^{18} T^{6} + 1728 p^{27} T^{7} + p^{36} T^{8}
7C2S4C_2 \wr S_4 14840T+9079372pT29122761384p2T3+9758964239914p3T49122761384p11T5+9079372p19T64840p27T7+p36T8 1 - 4840 T + 9079372 p T^{2} - 9122761384 p^{2} T^{3} + 9758964239914 p^{3} T^{4} - 9122761384 p^{11} T^{5} + 9079372 p^{19} T^{6} - 4840 p^{27} T^{7} + p^{36} T^{8}
11C2S4C_2 \wr S_4 1+15824T+3272030348T2+110832700706512T3+11168899925921994134T4+110832700706512p9T5+3272030348p18T6+15824p27T7+p36T8 1 + 15824 T + 3272030348 T^{2} + 110832700706512 T^{3} + 11168899925921994134 T^{4} + 110832700706512 p^{9} T^{5} + 3272030348 p^{18} T^{6} + 15824 p^{27} T^{7} + p^{36} T^{8}
13C2S4C_2 \wr S_4 1+82440T+32507843788T2+2806427918898776T3+ 1 + 82440 T + 32507843788 T^{2} + 2806427918898776 T^{3} + 46 ⁣ ⁣7446\!\cdots\!74T4+2806427918898776p9T5+32507843788p18T6+82440p27T7+p36T8 T^{4} + 2806427918898776 p^{9} T^{5} + 32507843788 p^{18} T^{6} + 82440 p^{27} T^{7} + p^{36} T^{8}
17C2S4C_2 \wr S_4 1165912T+402742640540T257573640941296168T3+ 1 - 165912 T + 402742640540 T^{2} - 57573640941296168 T^{3} + 68 ⁣ ⁣9068\!\cdots\!90T457573640941296168p9T5+402742640540p18T6165912p27T7+p36T8 T^{4} - 57573640941296168 p^{9} T^{5} + 402742640540 p^{18} T^{6} - 165912 p^{27} T^{7} + p^{36} T^{8}
19C2S4C_2 \wr S_4 1+28416pT+878148739660T2+411395194240546048T3+ 1 + 28416 p T + 878148739660 T^{2} + 411395194240546048 T^{3} + 35 ⁣ ⁣0235\!\cdots\!02T4+411395194240546048p9T5+878148739660p18T6+28416p28T7+p36T8 T^{4} + 411395194240546048 p^{9} T^{5} + 878148739660 p^{18} T^{6} + 28416 p^{28} T^{7} + p^{36} T^{8}
23C2S4C_2 \wr S_4 1729680T+3870147116348T23090843717350364304T3+ 1 - 729680 T + 3870147116348 T^{2} - 3090843717350364304 T^{3} + 76 ⁣ ⁣5476\!\cdots\!54T43090843717350364304p9T5+3870147116348p18T6729680p27T7+p36T8 T^{4} - 3090843717350364304 p^{9} T^{5} + 3870147116348 p^{18} T^{6} - 729680 p^{27} T^{7} + p^{36} T^{8}
29C2S4C_2 \wr S_4 1+1850864T+51896421352052T2+76884864756743058576T3+ 1 + 1850864 T + 51896421352052 T^{2} + 76884864756743058576 T^{3} + 10 ⁣ ⁣4210\!\cdots\!42T4+76884864756743058576p9T5+51896421352052p18T6+1850864p27T7+p36T8 T^{4} + 76884864756743058576 p^{9} T^{5} + 51896421352052 p^{18} T^{6} + 1850864 p^{27} T^{7} + p^{36} T^{8}
31C2S4C_2 \wr S_4 1103160pT+80766069413044T2 1 - 103160 p T + 80766069413044 T^{2} - 19 ⁣ ⁣9219\!\cdots\!92T3+ T^{3} + 28 ⁣ ⁣7428\!\cdots\!74T4 T^{4} - 19 ⁣ ⁣9219\!\cdots\!92p9T5+80766069413044p18T6103160p28T7+p36T8 p^{9} T^{5} + 80766069413044 p^{18} T^{6} - 103160 p^{28} T^{7} + p^{36} T^{8}
37C2S4C_2 \wr S_4 1+4187992T+316603511258092T2+ 1 + 4187992 T + 316603511258092 T^{2} + 22 ⁣ ⁣7222\!\cdots\!72T3+ T^{3} + 48 ⁣ ⁣5048\!\cdots\!50T4+ T^{4} + 22 ⁣ ⁣7222\!\cdots\!72p9T5+316603511258092p18T6+4187992p27T7+p36T8 p^{9} T^{5} + 316603511258092 p^{18} T^{6} + 4187992 p^{27} T^{7} + p^{36} T^{8}
41C2S4C_2 \wr S_4 1227704T+815482715263292T2 1 - 227704 T + 815482715263292 T^{2} - 24 ⁣ ⁣8024\!\cdots\!80T3+ T^{3} + 33 ⁣ ⁣4633\!\cdots\!46T4 T^{4} - 24 ⁣ ⁣8024\!\cdots\!80p9T5+815482715263292p18T6227704p27T7+p36T8 p^{9} T^{5} + 815482715263292 p^{18} T^{6} - 227704 p^{27} T^{7} + p^{36} T^{8}
43C2S4C_2 \wr S_4 1+1600352T+1039425217732396T2 1 + 1600352 T + 1039425217732396 T^{2} - 25 ⁣ ⁣4025\!\cdots\!40T3+ T^{3} + 54 ⁣ ⁣1454\!\cdots\!14T4 T^{4} - 25 ⁣ ⁣4025\!\cdots\!40p9T5+1039425217732396p18T6+1600352p27T7+p36T8 p^{9} T^{5} + 1039425217732396 p^{18} T^{6} + 1600352 p^{27} T^{7} + p^{36} T^{8}
47C2S4C_2 \wr S_4 118053904T+898850271089564T2 1 - 18053904 T + 898850271089564 T^{2} - 31 ⁣ ⁣8031\!\cdots\!80T3+ T^{3} + 25 ⁣ ⁣5425\!\cdots\!54T4 T^{4} - 31 ⁣ ⁣8031\!\cdots\!80p9T5+898850271089564p18T618053904p27T7+p36T8 p^{9} T^{5} + 898850271089564 p^{18} T^{6} - 18053904 p^{27} T^{7} + p^{36} T^{8}
53C2S4C_2 \wr S_4 1+29418288T+8246475311053268T2+ 1 + 29418288 T + 8246475311053268 T^{2} + 75 ⁣ ⁣5275\!\cdots\!52T3+ T^{3} + 31 ⁣ ⁣1031\!\cdots\!10T4+ T^{4} + 75 ⁣ ⁣5275\!\cdots\!52p9T5+8246475311053268p18T6+29418288p27T7+p36T8 p^{9} T^{5} + 8246475311053268 p^{18} T^{6} + 29418288 p^{27} T^{7} + p^{36} T^{8}
59C2S4C_2 \wr S_4 1+38300048T+12480247782213068T2+ 1 + 38300048 T + 12480247782213068 T^{2} + 64 ⁣ ⁣6064\!\cdots\!60T3+ T^{3} + 18 ⁣ ⁣6618\!\cdots\!66T4+ T^{4} + 64 ⁣ ⁣6064\!\cdots\!60p9T5+12480247782213068p18T6+38300048p27T7+p36T8 p^{9} T^{5} + 12480247782213068 p^{18} T^{6} + 38300048 p^{27} T^{7} + p^{36} T^{8}
61C2S4C_2 \wr S_4 199764648T+35070113945959756T2 1 - 99764648 T + 35070113945959756 T^{2} - 25 ⁣ ⁣0825\!\cdots\!08T3+ T^{3} + 53 ⁣ ⁣5453\!\cdots\!54T4 T^{4} - 25 ⁣ ⁣0825\!\cdots\!08p9T5+35070113945959756p18T699764648p27T7+p36T8 p^{9} T^{5} + 35070113945959756 p^{18} T^{6} - 99764648 p^{27} T^{7} + p^{36} T^{8}
67C2S4C_2 \wr S_4 1183717008T+94982362218124012T2 1 - 183717008 T + 94982362218124012 T^{2} - 13 ⁣ ⁣3613\!\cdots\!36T3+ T^{3} + 56 ⁣ ⁣9856\!\cdots\!98pT4 p T^{4} - 13 ⁣ ⁣3613\!\cdots\!36p9T5+94982362218124012p18T6183717008p27T7+p36T8 p^{9} T^{5} + 94982362218124012 p^{18} T^{6} - 183717008 p^{27} T^{7} + p^{36} T^{8}
71C2S4C_2 \wr S_4 1181868080T+187559006549429756T2 1 - 181868080 T + 187559006549429756 T^{2} - 24 ⁣ ⁣8024\!\cdots\!80T3+ T^{3} + 12 ⁣ ⁣8612\!\cdots\!86T4 T^{4} - 24 ⁣ ⁣8024\!\cdots\!80p9T5+187559006549429756p18T6181868080p27T7+p36T8 p^{9} T^{5} + 187559006549429756 p^{18} T^{6} - 181868080 p^{27} T^{7} + p^{36} T^{8}
73C2S4C_2 \wr S_4 1254539160T+208692556022844412T2 1 - 254539160 T + 208692556022844412 T^{2} - 39 ⁣ ⁣6439\!\cdots\!64T3+ T^{3} + 17 ⁣ ⁣8217\!\cdots\!82T4 T^{4} - 39 ⁣ ⁣6439\!\cdots\!64p9T5+208692556022844412p18T6254539160p27T7+p36T8 p^{9} T^{5} + 208692556022844412 p^{18} T^{6} - 254539160 p^{27} T^{7} + p^{36} T^{8}
79C2S4C_2 \wr S_4 1831578184T+541079473633225972T2 1 - 831578184 T + 541079473633225972 T^{2} - 23 ⁣ ⁣6423\!\cdots\!64T3+ T^{3} + 93 ⁣ ⁣9093\!\cdots\!90T4 T^{4} - 23 ⁣ ⁣6423\!\cdots\!64p9T5+541079473633225972p18T6831578184p27T7+p36T8 p^{9} T^{5} + 541079473633225972 p^{18} T^{6} - 831578184 p^{27} T^{7} + p^{36} T^{8}
83C2S4C_2 \wr S_4 1687923952T+569668186782399404T2 1 - 687923952 T + 569668186782399404 T^{2} - 18 ⁣ ⁣0418\!\cdots\!04T3+ T^{3} + 11 ⁣ ⁣5011\!\cdots\!50T4 T^{4} - 18 ⁣ ⁣0418\!\cdots\!04p9T5+569668186782399404p18T6687923952p27T7+p36T8 p^{9} T^{5} + 569668186782399404 p^{18} T^{6} - 687923952 p^{27} T^{7} + p^{36} T^{8}
89C2S4C_2 \wr S_4 1627627272T+1209510335623979132T2 1 - 627627272 T + 1209510335623979132 T^{2} - 64 ⁣ ⁣4464\!\cdots\!44T3+ T^{3} + 60 ⁣ ⁣1460\!\cdots\!14T4 T^{4} - 64 ⁣ ⁣4464\!\cdots\!44p9T5+1209510335623979132p18T6627627272p27T7+p36T8 p^{9} T^{5} + 1209510335623979132 p^{18} T^{6} - 627627272 p^{27} T^{7} + p^{36} T^{8}
97C2S4C_2 \wr S_4 1+889385880T+757736193785874268T2+ 1 + 889385880 T + 757736193785874268 T^{2} + 68 ⁣ ⁣8068\!\cdots\!80T3+ T^{3} + 12 ⁣ ⁣3412\!\cdots\!34T4+ T^{4} + 68 ⁣ ⁣8068\!\cdots\!80p9T5+757736193785874268p18T6+889385880p27T7+p36T8 p^{9} T^{5} + 757736193785874268 p^{18} T^{6} + 889385880 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

−7.21953944298647234879238008663, −6.70977829897245237584896270327, −6.56811404743840202616116417650, −6.40366981143595671452677332712, −6.40069119132911164184641280463, −5.84300716679751072964329906050, −5.45509633397555028154597785826, −5.43389612054543715067250299516, −5.29653747536534842203075401286, −4.90532976385749784906739601087, −4.73795703653614007245062783312, −4.45795018179924614251708434492, −4.29563866172337722579707232297, −3.98139426579004466978176649471, −3.57644495740852728200879200734, −3.44447708796586852349358038108, −3.40393120353781581012110011707, −2.56856758471296464469941357079, −2.29251621766701040935145204151, −2.08432114502390631600488445208, −2.07997878355662713525033593925, −1.31408191973900606143495366254, −1.13015958075911418184787062208, −1.04489250663729980680634738429, −0.73616208536664821684434958386, 0, 0, 0, 0, 0.73616208536664821684434958386, 1.04489250663729980680634738429, 1.13015958075911418184787062208, 1.31408191973900606143495366254, 2.07997878355662713525033593925, 2.08432114502390631600488445208, 2.29251621766701040935145204151, 2.56856758471296464469941357079, 3.40393120353781581012110011707, 3.44447708796586852349358038108, 3.57644495740852728200879200734, 3.98139426579004466978176649471, 4.29563866172337722579707232297, 4.45795018179924614251708434492, 4.73795703653614007245062783312, 4.90532976385749784906739601087, 5.29653747536534842203075401286, 5.43389612054543715067250299516, 5.45509633397555028154597785826, 5.84300716679751072964329906050, 6.40069119132911164184641280463, 6.40366981143595671452677332712, 6.56811404743840202616116417650, 6.70977829897245237584896270327, 7.21953944298647234879238008663

Graph of the ZZ-function along the critical line