Properties

 Degree 20 Conductor $2^{30}$ Sign $1$ Motivic weight 13 Primitive no Self-dual yes Analytic rank 0

Origins of factors

Dirichlet series

 L(s)  = 1 + 110·2-s + 3.69e3·4-s + 5.86e5·7-s − 1.27e5·8-s + 8.97e6·9-s + 6.45e7·14-s − 1.15e6·16-s + 2.17e8·17-s + 9.87e8·18-s − 7.86e7·23-s + 4.56e9·25-s + 2.16e9·28-s + 6.48e8·31-s − 1.11e9·32-s + 2.39e10·34-s + 3.31e10·36-s + 5.93e10·41-s − 8.65e9·46-s − 1.01e10·47-s − 2.20e11·49-s + 5.02e11·50-s − 7.50e10·56-s + 7.13e10·62-s + 5.27e12·63-s − 5.49e11·64-s + 8.02e11·68-s + 7.26e11·71-s + ⋯
 L(s)  = 1 + 1.21·2-s + 0.450·4-s + 1.88·7-s − 0.172·8-s + 5.63·9-s + 2.29·14-s − 0.0172·16-s + 2.18·17-s + 6.84·18-s − 0.110·23-s + 3.73·25-s + 0.849·28-s + 0.131·31-s − 0.184·32-s + 2.65·34-s + 2.53·36-s + 1.95·41-s − 0.134·46-s − 0.137·47-s − 2.27·49-s + 4.54·50-s − 0.324·56-s + 0.159·62-s + 10.6·63-s − 0.999·64-s + 0.984·68-s + 0.672·71-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(14-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{30}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$20$$ $$N$$ = $$2^{30}$$ $$\varepsilon$$ = $1$ motivic weight = $$13$$ character : induced by $\chi_{8} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(20,\ 2^{30} ,\ ( \ : [13/2]^{10} ),\ 1 )$ $L(7)$ $\approx$ $137.509$ $L(\frac12)$ $\approx$ $137.509$ $L(\frac{15}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p(T)$$ is a polynomial of degree 20. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad2 $$1 - 55 p T + 1051 p^{3} T^{2} - 6109 p^{6} T^{3} - 457 p^{11} T^{4} + 27561 p^{17} T^{5} - 457 p^{24} T^{6} - 6109 p^{32} T^{7} + 1051 p^{42} T^{8} - 55 p^{53} T^{9} + p^{65} T^{10}$$
good3 $$1 - 8978642 T^{2} + 4658209641485 p^{2} T^{4} - 181750892080261208 p^{6} T^{6} + 64862993246405867138 p^{14} T^{8} -$$$$14\!\cdots\!28$$$$p^{18} T^{10} + 64862993246405867138 p^{40} T^{12} - 181750892080261208 p^{58} T^{14} + 4658209641485 p^{80} T^{16} - 8978642 p^{104} T^{18} + p^{130} T^{20}$$
5 $$1 - 4565314338 T^{2} + 10147469641592691461 T^{4} -$$$$57\!\cdots\!32$$$$p^{2} T^{6} +$$$$10\!\cdots\!74$$$$p^{6} T^{8} -$$$$18\!\cdots\!44$$$$p^{10} T^{10} +$$$$10\!\cdots\!74$$$$p^{32} T^{12} -$$$$57\!\cdots\!32$$$$p^{54} T^{14} + 10147469641592691461 p^{78} T^{16} - 4565314338 p^{104} T^{18} + p^{130} T^{20}$$
7 $$( 1 - 293480 T + 239610643203 T^{2} - 1947286877479264 p^{2} T^{3} +$$$$65\!\cdots\!26$$$$p^{2} T^{4} -$$$$39\!\cdots\!96$$$$p^{3} T^{5} +$$$$65\!\cdots\!26$$$$p^{15} T^{6} - 1947286877479264 p^{28} T^{7} + 239610643203 p^{39} T^{8} - 293480 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
11 $$1 - 1482155238098 p^{2} T^{2} +$$$$11\!\cdots\!01$$$$p^{4} T^{4} -$$$$59\!\cdots\!00$$$$p^{6} T^{6} +$$$$23\!\cdots\!38$$$$p^{8} T^{8} -$$$$74\!\cdots\!64$$$$p^{10} T^{10} +$$$$23\!\cdots\!38$$$$p^{34} T^{12} -$$$$59\!\cdots\!00$$$$p^{58} T^{14} +$$$$11\!\cdots\!01$$$$p^{82} T^{16} - 1482155238098 p^{106} T^{18} + p^{130} T^{20}$$
13 $$1 - 1881700477395890 T^{2} +$$$$17\!\cdots\!13$$$$T^{4} -$$$$10\!\cdots\!00$$$$T^{6} +$$$$48\!\cdots\!02$$$$T^{8} -$$$$16\!\cdots\!20$$$$T^{10} +$$$$48\!\cdots\!02$$$$p^{26} T^{12} -$$$$10\!\cdots\!00$$$$p^{52} T^{14} +$$$$17\!\cdots\!13$$$$p^{78} T^{16} - 1881700477395890 p^{104} T^{18} + p^{130} T^{20}$$
17 $$( 1 - 108663002 T + 32276400633367229 T^{2} -$$$$23\!\cdots\!84$$$$T^{3} +$$$$50\!\cdots\!42$$$$T^{4} -$$$$29\!\cdots\!76$$$$T^{5} +$$$$50\!\cdots\!42$$$$p^{13} T^{6} -$$$$23\!\cdots\!84$$$$p^{26} T^{7} + 32276400633367229 p^{39} T^{8} - 108663002 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
19 $$1 - 8567413845488966 p T^{2} +$$$$13\!\cdots\!97$$$$T^{4} -$$$$80\!\cdots\!36$$$$T^{6} +$$$$39\!\cdots\!66$$$$T^{8} -$$$$17\!\cdots\!68$$$$T^{10} +$$$$39\!\cdots\!66$$$$p^{26} T^{12} -$$$$80\!\cdots\!36$$$$p^{52} T^{14} +$$$$13\!\cdots\!97$$$$p^{78} T^{16} - 8567413845488966 p^{105} T^{18} + p^{130} T^{20}$$
23 $$( 1 + 39339976 T + 1242751119340586771 T^{2} +$$$$42\!\cdots\!92$$$$T^{3} +$$$$76\!\cdots\!02$$$$T^{4} +$$$$38\!\cdots\!08$$$$T^{5} +$$$$76\!\cdots\!02$$$$p^{13} T^{6} +$$$$42\!\cdots\!92$$$$p^{26} T^{7} + 1242751119340586771 p^{39} T^{8} + 39339976 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
29 $$1 - 60047671171855484498 T^{2} +$$$$18\!\cdots\!41$$$$T^{4} -$$$$38\!\cdots\!60$$$$T^{6} +$$$$58\!\cdots\!78$$$$T^{8} -$$$$67\!\cdots\!84$$$$T^{10} +$$$$58\!\cdots\!78$$$$p^{26} T^{12} -$$$$38\!\cdots\!60$$$$p^{52} T^{14} +$$$$18\!\cdots\!41$$$$p^{78} T^{16} - 60047671171855484498 p^{104} T^{18} + p^{130} T^{20}$$
31 $$( 1 - 324116896 T + 94759416869201467419 T^{2} -$$$$76\!\cdots\!28$$$$T^{3} +$$$$40\!\cdots\!58$$$$T^{4} -$$$$32\!\cdots\!48$$$$T^{5} +$$$$40\!\cdots\!58$$$$p^{13} T^{6} -$$$$76\!\cdots\!28$$$$p^{26} T^{7} + 94759416869201467419 p^{39} T^{8} - 324116896 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
37 $$1 -$$$$85\!\cdots\!14$$$$T^{2} +$$$$35\!\cdots\!29$$$$T^{4} -$$$$83\!\cdots\!24$$$$T^{6} +$$$$11\!\cdots\!78$$$$T^{8} -$$$$14\!\cdots\!64$$$$T^{10} +$$$$11\!\cdots\!78$$$$p^{26} T^{12} -$$$$83\!\cdots\!24$$$$p^{52} T^{14} +$$$$35\!\cdots\!29$$$$p^{78} T^{16} -$$$$85\!\cdots\!14$$$$p^{104} T^{18} + p^{130} T^{20}$$
41 $$( 1 - 29662320178 T +$$$$14\!\cdots\!89$$$$T^{2} -$$$$15\!\cdots\!16$$$$T^{3} +$$$$10\!\cdots\!62$$$$T^{4} -$$$$29\!\cdots\!76$$$$T^{5} +$$$$10\!\cdots\!62$$$$p^{13} T^{6} -$$$$15\!\cdots\!16$$$$p^{26} T^{7} +$$$$14\!\cdots\!89$$$$p^{39} T^{8} - 29662320178 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
43 $$1 -$$$$61\!\cdots\!14$$$$T^{2} +$$$$15\!\cdots\!05$$$$T^{4} -$$$$25\!\cdots\!04$$$$T^{6} +$$$$52\!\cdots\!62$$$$T^{8} -$$$$10\!\cdots\!24$$$$T^{10} +$$$$52\!\cdots\!62$$$$p^{26} T^{12} -$$$$25\!\cdots\!04$$$$p^{52} T^{14} +$$$$15\!\cdots\!05$$$$p^{78} T^{16} -$$$$61\!\cdots\!14$$$$p^{104} T^{18} + p^{130} T^{20}$$
47 $$( 1 + 5088267408 T +$$$$10\!\cdots\!43$$$$T^{2} +$$$$16\!\cdots\!28$$$$T^{3} +$$$$79\!\cdots\!70$$$$T^{4} +$$$$90\!\cdots\!04$$$$T^{5} +$$$$79\!\cdots\!70$$$$p^{13} T^{6} +$$$$16\!\cdots\!28$$$$p^{26} T^{7} +$$$$10\!\cdots\!43$$$$p^{39} T^{8} + 5088267408 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
53 $$1 -$$$$15\!\cdots\!98$$$$T^{2} +$$$$12\!\cdots\!49$$$$T^{4} -$$$$63\!\cdots\!52$$$$T^{6} +$$$$24\!\cdots\!06$$$$T^{8} -$$$$72\!\cdots\!20$$$$T^{10} +$$$$24\!\cdots\!06$$$$p^{26} T^{12} -$$$$63\!\cdots\!52$$$$p^{52} T^{14} +$$$$12\!\cdots\!49$$$$p^{78} T^{16} -$$$$15\!\cdots\!98$$$$p^{104} T^{18} + p^{130} T^{20}$$
59 $$1 -$$$$54\!\cdots\!82$$$$T^{2} +$$$$16\!\cdots\!77$$$$T^{4} -$$$$32\!\cdots\!52$$$$T^{6} +$$$$49\!\cdots\!86$$$$T^{8} -$$$$58\!\cdots\!80$$$$T^{10} +$$$$49\!\cdots\!86$$$$p^{26} T^{12} -$$$$32\!\cdots\!52$$$$p^{52} T^{14} +$$$$16\!\cdots\!77$$$$p^{78} T^{16} -$$$$54\!\cdots\!82$$$$p^{104} T^{18} + p^{130} T^{20}$$
61 $$1 -$$$$10\!\cdots\!62$$$$T^{2} +$$$$47\!\cdots\!37$$$$T^{4} -$$$$13\!\cdots\!12$$$$T^{6} +$$$$29\!\cdots\!66$$$$T^{8} -$$$$51\!\cdots\!60$$$$T^{10} +$$$$29\!\cdots\!66$$$$p^{26} T^{12} -$$$$13\!\cdots\!12$$$$p^{52} T^{14} +$$$$47\!\cdots\!37$$$$p^{78} T^{16} -$$$$10\!\cdots\!62$$$$p^{104} T^{18} + p^{130} T^{20}$$
67 $$1 -$$$$19\!\cdots\!42$$$$T^{2} +$$$$23\!\cdots\!49$$$$T^{4} -$$$$18\!\cdots\!88$$$$T^{6} +$$$$12\!\cdots\!86$$$$T^{8} -$$$$71\!\cdots\!60$$$$T^{10} +$$$$12\!\cdots\!86$$$$p^{26} T^{12} -$$$$18\!\cdots\!88$$$$p^{52} T^{14} +$$$$23\!\cdots\!49$$$$p^{78} T^{16} -$$$$19\!\cdots\!42$$$$p^{104} T^{18} + p^{130} T^{20}$$
71 $$( 1 - 363180589992 T +$$$$34\!\cdots\!95$$$$T^{2} -$$$$14\!\cdots\!16$$$$T^{3} +$$$$61\!\cdots\!94$$$$T^{4} -$$$$22\!\cdots\!64$$$$T^{5} +$$$$61\!\cdots\!94$$$$p^{13} T^{6} -$$$$14\!\cdots\!16$$$$p^{26} T^{7} +$$$$34\!\cdots\!95$$$$p^{39} T^{8} - 363180589992 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
73 $$( 1 + 316620182766 T +$$$$64\!\cdots\!53$$$$T^{2} +$$$$24\!\cdots\!84$$$$T^{3} +$$$$18\!\cdots\!54$$$$T^{4} +$$$$60\!\cdots\!52$$$$T^{5} +$$$$18\!\cdots\!54$$$$p^{13} T^{6} +$$$$24\!\cdots\!84$$$$p^{26} T^{7} +$$$$64\!\cdots\!53$$$$p^{39} T^{8} + 316620182766 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
79 $$( 1 - 2722551782672 T +$$$$12\!\cdots\!19$$$$T^{2} -$$$$22\!\cdots\!64$$$$T^{3} +$$$$55\!\cdots\!14$$$$T^{4} -$$$$94\!\cdots\!68$$$$T^{5} +$$$$55\!\cdots\!14$$$$p^{13} T^{6} -$$$$22\!\cdots\!64$$$$p^{26} T^{7} +$$$$12\!\cdots\!19$$$$p^{39} T^{8} - 2722551782672 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
83 $$1 -$$$$43\!\cdots\!74$$$$T^{2} +$$$$87\!\cdots\!65$$$$T^{4} -$$$$12\!\cdots\!64$$$$T^{6} +$$$$14\!\cdots\!62$$$$T^{8} -$$$$14\!\cdots\!44$$$$T^{10} +$$$$14\!\cdots\!62$$$$p^{26} T^{12} -$$$$12\!\cdots\!64$$$$p^{52} T^{14} +$$$$87\!\cdots\!65$$$$p^{78} T^{16} -$$$$43\!\cdots\!74$$$$p^{104} T^{18} + p^{130} T^{20}$$
89 $$( 1 - 2753172404002 T +$$$$62\!\cdots\!29$$$$T^{2} -$$$$14\!\cdots\!84$$$$T^{3} +$$$$21\!\cdots\!54$$$$T^{4} -$$$$47\!\cdots\!48$$$$T^{5} +$$$$21\!\cdots\!54$$$$p^{13} T^{6} -$$$$14\!\cdots\!84$$$$p^{26} T^{7} +$$$$62\!\cdots\!29$$$$p^{39} T^{8} - 2753172404002 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
97 $$( 1 - 680566660394 T +$$$$17\!\cdots\!53$$$$T^{2} +$$$$11\!\cdots\!08$$$$T^{3} +$$$$15\!\cdots\!70$$$$T^{4} +$$$$20\!\cdots\!28$$$$T^{5} +$$$$15\!\cdots\!70$$$$p^{13} T^{6} +$$$$11\!\cdots\!08$$$$p^{26} T^{7} +$$$$17\!\cdots\!53$$$$p^{39} T^{8} - 680566660394 p^{52} T^{9} + p^{65} T^{10} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}