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Properties

Label 8-1083e4-1.1-c3e4-0-2
Degree 88
Conductor 1.376×10121.376\times 10^{12}
Sign 11
Analytic cond. 1.66716×1071.66716\times 10^{7}
Root an. cond. 7.993687.99368
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 12·3-s − 3·4-s + 18·5-s − 24·6-s − 10·7-s − 4·8-s + 90·9-s − 36·10-s − 14·11-s − 36·12-s − 56·13-s + 20·14-s + 216·15-s + 39·16-s − 186·17-s − 180·18-s − 54·20-s − 120·21-s + 28·22-s − 84·23-s − 48·24-s − 173·25-s + 112·26-s + 540·27-s + 30·28-s + 236·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 2.30·3-s − 3/8·4-s + 1.60·5-s − 1.63·6-s − 0.539·7-s − 0.176·8-s + 10/3·9-s − 1.13·10-s − 0.383·11-s − 0.866·12-s − 1.19·13-s + 0.381·14-s + 3.71·15-s + 0.609·16-s − 2.65·17-s − 2.35·18-s − 0.603·20-s − 1.24·21-s + 0.271·22-s − 0.761·23-s − 0.408·24-s − 1.38·25-s + 0.844·26-s + 3.84·27-s + 0.202·28-s + 1.51·29-s + ⋯

Functional equation

Λ(s)=((34⋅198)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((34⋅198)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅1983^{4} \cdot 19^{8}
Sign: 11
Analytic conductor: 1.66716×1071.66716\times 10^{7}
Root analytic conductor: 7.993687.99368
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 34⋅198, ( :3/2,3/2,3/2,3/2), 1)(8,\ 3^{4} \cdot 19^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3C1C_1 (1−pT)4 ( 1 - p T )^{4}
19 1 1
good2C2≀S4C_2 \wr S_4 1+pT+7T2+3p3T3+19pT4+3p6T5+7p6T6+p10T7+p12T8 1 + p T + 7 T^{2} + 3 p^{3} T^{3} + 19 p T^{4} + 3 p^{6} T^{5} + 7 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8}
5C2≀S4C_2 \wr S_4 1−18T+497T2−5634T3+89156T4−5634p3T5+497p6T6−18p9T7+p12T8 1 - 18 T + 497 T^{2} - 5634 T^{3} + 89156 T^{4} - 5634 p^{3} T^{5} + 497 p^{6} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8}
7C2≀S4C_2 \wr S_4 1+10T+957T2+11898T3+419908T4+11898p3T5+957p6T6+10p9T7+p12T8 1 + 10 T + 957 T^{2} + 11898 T^{3} + 419908 T^{4} + 11898 p^{3} T^{5} + 957 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8}
11C2≀S4C_2 \wr S_4 1+14T+2049T2+16954T3+3844088T4+16954p3T5+2049p6T6+14p9T7+p12T8 1 + 14 T + 2049 T^{2} + 16954 T^{3} + 3844088 T^{4} + 16954 p^{3} T^{5} + 2049 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8}
13C2≀S4C_2 \wr S_4 1+56T+7388T2+271368T3+22128422T4+271368p3T5+7388p6T6+56p9T7+p12T8 1 + 56 T + 7388 T^{2} + 271368 T^{3} + 22128422 T^{4} + 271368 p^{3} T^{5} + 7388 p^{6} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8}
17C2≀S4C_2 \wr S_4 1+186T+89p2T2+2469834T3+193997780T4+2469834p3T5+89p8T6+186p9T7+p12T8 1 + 186 T + 89 p^{2} T^{2} + 2469834 T^{3} + 193997780 T^{4} + 2469834 p^{3} T^{5} + 89 p^{8} T^{6} + 186 p^{9} T^{7} + p^{12} T^{8}
23C2≀S4C_2 \wr S_4 1+84T+30344T2+1222788T3+17501794pT4+1222788p3T5+30344p6T6+84p9T7+p12T8 1 + 84 T + 30344 T^{2} + 1222788 T^{3} + 17501794 p T^{4} + 1222788 p^{3} T^{5} + 30344 p^{6} T^{6} + 84 p^{9} T^{7} + p^{12} T^{8}
29C2≀S4C_2 \wr S_4 1−236T+38296T2−2172756T3+400254782T4−2172756p3T5+38296p6T6−236p9T7+p12T8 1 - 236 T + 38296 T^{2} - 2172756 T^{3} + 400254782 T^{4} - 2172756 p^{3} T^{5} + 38296 p^{6} T^{6} - 236 p^{9} T^{7} + p^{12} T^{8}
31C2≀S4C_2 \wr S_4 1+112T+76212T2+6302064T3+2836990870T4+6302064p3T5+76212p6T6+112p9T7+p12T8 1 + 112 T + 76212 T^{2} + 6302064 T^{3} + 2836990870 T^{4} + 6302064 p^{3} T^{5} + 76212 p^{6} T^{6} + 112 p^{9} T^{7} + p^{12} T^{8}
37C2≀S4C_2 \wr S_4 1+544T+224364T2+1677600pT3+16007180134T4+1677600p4T5+224364p6T6+544p9T7+p12T8 1 + 544 T + 224364 T^{2} + 1677600 p T^{3} + 16007180134 T^{4} + 1677600 p^{4} T^{5} + 224364 p^{6} T^{6} + 544 p^{9} T^{7} + p^{12} T^{8}
41C2≀S4C_2 \wr S_4 1−68T+159832T2−24974988T3+12851495438T4−24974988p3T5+159832p6T6−68p9T7+p12T8 1 - 68 T + 159832 T^{2} - 24974988 T^{3} + 12851495438 T^{4} - 24974988 p^{3} T^{5} + 159832 p^{6} T^{6} - 68 p^{9} T^{7} + p^{12} T^{8}
43C2≀S4C_2 \wr S_4 1+674T+355397T2+126100458T3+39633391484T4+126100458p3T5+355397p6T6+674p9T7+p12T8 1 + 674 T + 355397 T^{2} + 126100458 T^{3} + 39633391484 T^{4} + 126100458 p^{3} T^{5} + 355397 p^{6} T^{6} + 674 p^{9} T^{7} + p^{12} T^{8}
47C2≀S4C_2 \wr S_4 1+114T+171337T2−6705570T3+18249717032T4−6705570p3T5+171337p6T6+114p9T7+p12T8 1 + 114 T + 171337 T^{2} - 6705570 T^{3} + 18249717032 T^{4} - 6705570 p^{3} T^{5} + 171337 p^{6} T^{6} + 114 p^{9} T^{7} + p^{12} T^{8}
53C2≀S4C_2 \wr S_4 1+540T+372744T2+73265892T3+44247102910T4+73265892p3T5+372744p6T6+540p9T7+p12T8 1 + 540 T + 372744 T^{2} + 73265892 T^{3} + 44247102910 T^{4} + 73265892 p^{3} T^{5} + 372744 p^{6} T^{6} + 540 p^{9} T^{7} + p^{12} T^{8}
59C2≀S4C_2 \wr S_4 1+582764T2+677376T3+165505100406T4+677376p3T5+582764p6T6+p12T8 1 + 582764 T^{2} + 677376 T^{3} + 165505100406 T^{4} + 677376 p^{3} T^{5} + 582764 p^{6} T^{6} + p^{12} T^{8}
61C2≀S4C_2 \wr S_4 1+558T+87745T2−60123114T3−12563228676T4−60123114p3T5+87745p6T6+558p9T7+p12T8 1 + 558 T + 87745 T^{2} - 60123114 T^{3} - 12563228676 T^{4} - 60123114 p^{3} T^{5} + 87745 p^{6} T^{6} + 558 p^{9} T^{7} + p^{12} T^{8}
67C2≀S4C_2 \wr S_4 1−320T+567788T2−76531008T3+195546355478T4−76531008p3T5+567788p6T6−320p9T7+p12T8 1 - 320 T + 567788 T^{2} - 76531008 T^{3} + 195546355478 T^{4} - 76531008 p^{3} T^{5} + 567788 p^{6} T^{6} - 320 p^{9} T^{7} + p^{12} T^{8}
71C2≀S4C_2 \wr S_4 1+872T+1195948T2+855329544T3+617068843718T4+855329544p3T5+1195948p6T6+872p9T7+p12T8 1 + 872 T + 1195948 T^{2} + 855329544 T^{3} + 617068843718 T^{4} + 855329544 p^{3} T^{5} + 1195948 p^{6} T^{6} + 872 p^{9} T^{7} + p^{12} T^{8}
73C2≀S4C_2 \wr S_4 1−446T+949913T2−497765814T3+450278879396T4−497765814p3T5+949913p6T6−446p9T7+p12T8 1 - 446 T + 949913 T^{2} - 497765814 T^{3} + 450278879396 T^{4} - 497765814 p^{3} T^{5} + 949913 p^{6} T^{6} - 446 p^{9} T^{7} + p^{12} T^{8}
79C2≀S4C_2 \wr S_4 1−1232T+1638900T2−1309543440T3+1185149213782T4−1309543440p3T5+1638900p6T6−1232p9T7+p12T8 1 - 1232 T + 1638900 T^{2} - 1309543440 T^{3} + 1185149213782 T^{4} - 1309543440 p^{3} T^{5} + 1638900 p^{6} T^{6} - 1232 p^{9} T^{7} + p^{12} T^{8}
83C2≀S4C_2 \wr S_4 1+780T+1378672T2+750270540T3+1034898364334T4+750270540p3T5+1378672p6T6+780p9T7+p12T8 1 + 780 T + 1378672 T^{2} + 750270540 T^{3} + 1034898364334 T^{4} + 750270540 p^{3} T^{5} + 1378672 p^{6} T^{6} + 780 p^{9} T^{7} + p^{12} T^{8}
89C2≀S4C_2 \wr S_4 1+60T+670872T2+799448436T3−22570573490T4+799448436p3T5+670872p6T6+60p9T7+p12T8 1 + 60 T + 670872 T^{2} + 799448436 T^{3} - 22570573490 T^{4} + 799448436 p^{3} T^{5} + 670872 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8}
97C2≀S4C_2 \wr S_4 1−2224T+3918308T2−4720818960T3+5246860342598T4−4720818960p3T5+3918308p6T6−2224p9T7+p12T8 1 - 2224 T + 3918308 T^{2} - 4720818960 T^{3} + 5246860342598 T^{4} - 4720818960 p^{3} T^{5} + 3918308 p^{6} T^{6} - 2224 p^{9} T^{7} + p^{12} 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

−6.99444767617302964160401139723, −6.78880942880939623954790292796, −6.78544317402228197397232799293, −6.47369784806651124658929465325, −6.36346002547948805656316049875, −6.34842099289411731514810344484, −5.61852994564036756428898907069, −5.57216143999279570689579074591, −5.30308179764151773959137744652, −5.06933111729948751757873268485, −4.71292160729323287850402362016, −4.65936687596261913338138112576, −4.30230345622050608631866897412, −3.98013480037129183245341730098, −3.69339932173430345584360085638, −3.62542530743543034024802280781, −3.16182031585906213137865382418, −2.95894232225769712613865729955, −2.78833382287160930987867901153, −2.23475035145670986801937773515, −2.22194112583618831754088508531, −2.16145006169813327280591812159, −1.58843449863729883963914821559, −1.41068531544588663315762371292, −1.39488154571977503729677050011, 0, 0, 0, 0, 1.39488154571977503729677050011, 1.41068531544588663315762371292, 1.58843449863729883963914821559, 2.16145006169813327280591812159, 2.22194112583618831754088508531, 2.23475035145670986801937773515, 2.78833382287160930987867901153, 2.95894232225769712613865729955, 3.16182031585906213137865382418, 3.62542530743543034024802280781, 3.69339932173430345584360085638, 3.98013480037129183245341730098, 4.30230345622050608631866897412, 4.65936687596261913338138112576, 4.71292160729323287850402362016, 5.06933111729948751757873268485, 5.30308179764151773959137744652, 5.57216143999279570689579074591, 5.61852994564036756428898907069, 6.34842099289411731514810344484, 6.36346002547948805656316049875, 6.47369784806651124658929465325, 6.78544317402228197397232799293, 6.78880942880939623954790292796, 6.99444767617302964160401139723

Graph of the ZZ-function along the critical line