# Properties

 Label 16-304e8-1.1-c6e8-0-0 Degree $16$ Conductor $7.294\times 10^{19}$ Sign $1$ Analytic cond. $5.72305\times 10^{14}$ Root an. cond. $8.36280$ Motivic weight $6$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·5-s − 362·7-s + 742·9-s − 902·11-s + 1.55e3·17-s − 6.23e3·19-s + 1.88e4·23-s − 6.85e4·25-s − 724·35-s + 3.35e5·43-s + 1.48e3·45-s + 5.70e5·47-s − 1.80e5·49-s − 1.80e3·55-s − 6.32e5·61-s − 2.68e5·63-s − 8.52e5·73-s + 3.26e5·77-s − 9.72e5·81-s − 4.41e5·83-s + 3.10e3·85-s − 1.24e4·95-s − 6.69e5·99-s − 1.81e6·101-s + 3.76e4·115-s − 5.61e5·119-s − 5.29e6·121-s + ⋯
 L(s)  = 1 + 0.0159·5-s − 1.05·7-s + 1.01·9-s − 0.677·11-s + 0.315·17-s − 0.908·19-s + 1.54·23-s − 4.38·25-s − 0.0168·35-s + 4.21·43-s + 0.0162·45-s + 5.49·47-s − 1.53·49-s − 0.0108·55-s − 2.78·61-s − 1.07·63-s − 2.19·73-s + 0.715·77-s − 1.82·81-s − 0.771·83-s + 0.00504·85-s − 0.0145·95-s − 0.689·99-s − 1.76·101-s + 0.0247·115-s − 0.332·119-s − 2.98·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{32} \cdot 19^{8}$$ Sign: $1$ Analytic conductor: $$5.72305\times 10^{14}$$ Root analytic conductor: $$8.36280$$ Motivic weight: $$6$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{304} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{32} \cdot 19^{8} ,\ ( \ : [3]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.001809907527$$ $$L(\frac12)$$ $$\approx$$ $$0.001809907527$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + 328 p T + 7370512 p T^{2} + 1950884072 p^{2} T^{3} + 70846142174 p^{4} T^{4} + 1950884072 p^{8} T^{5} + 7370512 p^{13} T^{6} + 328 p^{19} T^{7} + p^{24} T^{8}$$
good3 $$1 - 742 T^{2} + 507523 p T^{4} - 139764998 p^{2} T^{6} + 39023898956 p^{3} T^{8} - 139764998 p^{14} T^{10} + 507523 p^{25} T^{12} - 742 p^{36} T^{14} + p^{48} T^{16}$$
5 $$( 1 - T + 6858 p T^{2} + 57133 p^{2} T^{3} + 4953194 p^{3} T^{4} + 57133 p^{8} T^{5} + 6858 p^{13} T^{6} - p^{18} T^{7} + p^{24} T^{8} )^{2}$$
7 $$( 1 + 181 T + 139633 T^{2} - 29351086 T^{3} - 320349574 T^{4} - 29351086 p^{6} T^{5} + 139633 p^{12} T^{6} + 181 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
11 $$( 1 + 41 p T + 2951358 T^{2} + 3849343493 T^{3} + 4110119555458 T^{4} + 3849343493 p^{6} T^{5} + 2951358 p^{12} T^{6} + 41 p^{19} T^{7} + p^{24} T^{8} )^{2}$$
13 $$1 - 18101774 T^{2} + 149249945971081 T^{4} -$$$$86\!\cdots\!26$$$$T^{6} +$$$$43\!\cdots\!16$$$$T^{8} -$$$$86\!\cdots\!26$$$$p^{12} T^{10} + 149249945971081 p^{24} T^{12} - 18101774 p^{36} T^{14} + p^{48} T^{16}$$
17 $$( 1 - 775 T + 76306401 T^{2} - 62323644578 T^{3} + 2574852969478894 T^{4} - 62323644578 p^{6} T^{5} + 76306401 p^{12} T^{6} - 775 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
23 $$( 1 - 9410 T + 537866745 T^{2} - 3615134698390 T^{3} + 115088224709036668 T^{4} - 3615134698390 p^{6} T^{5} + 537866745 p^{12} T^{6} - 9410 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
29 $$1 - 1955972438 T^{2} + 1491101576657216569 T^{4} -$$$$39\!\cdots\!02$$$$T^{6} -$$$$19\!\cdots\!00$$$$T^{8} -$$$$39\!\cdots\!02$$$$p^{12} T^{10} + 1491101576657216569 p^{24} T^{12} - 1955972438 p^{36} T^{14} + p^{48} T^{16}$$
31 $$1 - 1360240952 T^{2} + 2738159624622996748 T^{4} -$$$$21\!\cdots\!68$$$$T^{6} +$$$$27\!\cdots\!82$$$$T^{8} -$$$$21\!\cdots\!68$$$$p^{12} T^{10} + 2738159624622996748 p^{24} T^{12} - 1360240952 p^{36} T^{14} + p^{48} T^{16}$$
37 $$1 - 7244329016 T^{2} + 23581771984582865740 T^{4} -$$$$37\!\cdots\!88$$$$T^{6} +$$$$48\!\cdots\!10$$$$T^{8} -$$$$37\!\cdots\!88$$$$p^{12} T^{10} + 23581771984582865740 p^{24} T^{12} - 7244329016 p^{36} T^{14} + p^{48} T^{16}$$
41 $$1 - 4812538424 T^{2} + 68132340249279236236 T^{4} -$$$$21\!\cdots\!16$$$$T^{6} +$$$$48\!\cdots\!26$$$$p T^{8} -$$$$21\!\cdots\!16$$$$p^{12} T^{10} + 68132340249279236236 p^{24} T^{12} - 4812538424 p^{36} T^{14} + p^{48} T^{16}$$
43 $$( 1 - 167521 T + 28059854686 T^{2} - 2860511677640443 T^{3} +$$$$27\!\cdots\!70$$$$T^{4} - 2860511677640443 p^{6} T^{5} + 28059854686 p^{12} T^{6} - 167521 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
47 $$( 1 - 285197 T + 53014565646 T^{2} - 6790341576595363 T^{3} +$$$$75\!\cdots\!74$$$$T^{4} - 6790341576595363 p^{6} T^{5} + 53014565646 p^{12} T^{6} - 285197 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
53 $$1 - 106845552662 T^{2} +$$$$57\!\cdots\!21$$$$T^{4} -$$$$20\!\cdots\!82$$$$T^{6} +$$$$53\!\cdots\!08$$$$T^{8} -$$$$20\!\cdots\!82$$$$p^{12} T^{10} +$$$$57\!\cdots\!21$$$$p^{24} T^{12} - 106845552662 p^{36} T^{14} + p^{48} T^{16}$$
59 $$1 - 283261383806 T^{2} +$$$$36\!\cdots\!69$$$$T^{4} -$$$$28\!\cdots\!06$$$$T^{6} +$$$$14\!\cdots\!44$$$$T^{8} -$$$$28\!\cdots\!06$$$$p^{12} T^{10} +$$$$36\!\cdots\!69$$$$p^{24} T^{12} - 283261383806 p^{36} T^{14} + p^{48} T^{16}$$
61 $$( 1 + 316007 T + 182153576986 T^{2} + 45019267728697357 T^{3} +$$$$13\!\cdots\!70$$$$T^{4} + 45019267728697357 p^{6} T^{5} + 182153576986 p^{12} T^{6} + 316007 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
67 $$1 - 284119902326 T^{2} +$$$$40\!\cdots\!01$$$$T^{4} -$$$$44\!\cdots\!94$$$$T^{6} +$$$$42\!\cdots\!76$$$$T^{8} -$$$$44\!\cdots\!94$$$$p^{12} T^{10} +$$$$40\!\cdots\!01$$$$p^{24} T^{12} - 284119902326 p^{36} T^{14} + p^{48} T^{16}$$
71 $$1 - 550447281824 T^{2} +$$$$16\!\cdots\!32$$$$T^{4} -$$$$34\!\cdots\!36$$$$T^{6} +$$$$52\!\cdots\!54$$$$T^{8} -$$$$34\!\cdots\!36$$$$p^{12} T^{10} +$$$$16\!\cdots\!32$$$$p^{24} T^{12} - 550447281824 p^{36} T^{14} + p^{48} T^{16}$$
73 $$( 1 + 426469 T + 412154349109 T^{2} + 175844843253819082 T^{3} +$$$$80\!\cdots\!22$$$$T^{4} + 175844843253819082 p^{6} T^{5} + 412154349109 p^{12} T^{6} + 426469 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
79 $$1 - 935745384632 T^{2} +$$$$48\!\cdots\!64$$$$T^{4} -$$$$17\!\cdots\!52$$$$T^{6} +$$$$48\!\cdots\!78$$$$T^{8} -$$$$17\!\cdots\!52$$$$p^{12} T^{10} +$$$$48\!\cdots\!64$$$$p^{24} T^{12} - 935745384632 p^{36} T^{14} + p^{48} T^{16}$$
83 $$( 1 + 220600 T + 1020667843728 T^{2} + 182031252335003816 T^{3} +$$$$45\!\cdots\!30$$$$T^{4} + 182031252335003816 p^{6} T^{5} + 1020667843728 p^{12} T^{6} + 220600 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
89 $$1 - 1390950826016 T^{2} +$$$$76\!\cdots\!16$$$$T^{4} -$$$$73\!\cdots\!44$$$$T^{6} -$$$$68\!\cdots\!94$$$$T^{8} -$$$$73\!\cdots\!44$$$$p^{12} T^{10} +$$$$76\!\cdots\!16$$$$p^{24} T^{12} - 1390950826016 p^{36} T^{14} + p^{48} T^{16}$$
97 $$1 - 5125637527808 T^{2} +$$$$12\!\cdots\!48$$$$T^{4} -$$$$18\!\cdots\!72$$$$T^{6} +$$$$18\!\cdots\!02$$$$T^{8} -$$$$18\!\cdots\!72$$$$p^{12} T^{10} +$$$$12\!\cdots\!48$$$$p^{24} T^{12} - 5125637527808 p^{36} T^{14} + p^{48} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$