# Properties

 Degree $8$ Conductor $2085136$ Sign $1$ Motivic weight $9$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 64·2-s + 84·3-s + 2.56e3·4-s − 1.39e3·5-s − 5.37e3·6-s + 1.23e4·7-s − 8.19e4·8-s − 2.75e4·9-s + 8.92e4·10-s − 1.04e5·11-s + 2.15e5·12-s + 1.20e5·13-s − 7.87e5·14-s − 1.17e5·15-s + 2.29e6·16-s − 4.12e5·17-s + 1.76e6·18-s + 5.21e5·19-s − 3.57e6·20-s + 1.03e6·21-s + 6.67e6·22-s + 3.01e6·23-s − 6.88e6·24-s + 1.94e6·25-s − 7.71e6·26-s + 1.06e6·27-s + 3.15e7·28-s + ⋯
 L(s)  = 1 − 2.82·2-s + 0.598·3-s + 5·4-s − 0.998·5-s − 1.69·6-s + 1.93·7-s − 7.07·8-s − 1.40·9-s + 2.82·10-s − 2.14·11-s + 2.99·12-s + 1.17·13-s − 5.47·14-s − 0.597·15-s + 35/4·16-s − 1.19·17-s + 3.96·18-s + 0.917·19-s − 4.99·20-s + 1.15·21-s + 6.07·22-s + 2.24·23-s − 4.23·24-s + 0.996·25-s − 3.30·26-s + 0.385·27-s + 9.68·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2085136$$    =    $$2^{4} \cdot 19^{4}$$ Sign: $1$ Motivic weight: $$9$$ Character: induced by $\chi_{38} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2085136,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$1.15634$$ $$L(\frac12)$$ $$\approx$$ $$1.15634$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$( 1 + p^{4} T )^{4}$$
19$C_1$ $$( 1 - p^{4} T )^{4}$$
good3$C_2 \wr S_4$ $$1 - 28 p T + 34625 T^{2} - 2096378 p T^{3} + 72317420 p^{2} T^{4} - 2096378 p^{10} T^{5} + 34625 p^{18} T^{6} - 28 p^{28} T^{7} + p^{36} T^{8}$$
5$C_2 \wr S_4$ $$1 + 279 p T - 206 p T^{2} + 65059389 p^{2} T^{3} + 62045170554 p^{3} T^{4} + 65059389 p^{11} T^{5} - 206 p^{19} T^{6} + 279 p^{28} T^{7} + p^{36} T^{8}$$
7$C_2 \wr S_4$ $$1 - 12307 T + 146953147 T^{2} - 177878821772 p T^{3} + 182915235883424 p^{2} T^{4} - 177878821772 p^{10} T^{5} + 146953147 p^{18} T^{6} - 12307 p^{27} T^{7} + p^{36} T^{8}$$
11$C_2 \wr S_4$ $$1 + 104249 T + 12830631428 T^{2} + 70344030474447 p T^{3} + 49503846191918961558 T^{4} + 70344030474447 p^{10} T^{5} + 12830631428 p^{18} T^{6} + 104249 p^{27} T^{7} + p^{36} T^{8}$$
13$C_2 \wr S_4$ $$1 - 120486 T + 11611428199 T^{2} - 1537985136135464 T^{3} +$$$$26\!\cdots\!96$$$$T^{4} - 1537985136135464 p^{9} T^{5} + 11611428199 p^{18} T^{6} - 120486 p^{27} T^{7} + p^{36} T^{8}$$
17$C_2 \wr S_4$ $$1 + 412139 T + 99978579665 T^{2} + 47122140457475886 T^{3} +$$$$22\!\cdots\!54$$$$T^{4} + 47122140457475886 p^{9} T^{5} + 99978579665 p^{18} T^{6} + 412139 p^{27} T^{7} + p^{36} T^{8}$$
23$C_2 \wr S_4$ $$1 - 3010300 T + 8067019202447 T^{2} - 26728667320812612 p^{2} T^{3} +$$$$22\!\cdots\!92$$$$T^{4} - 26728667320812612 p^{11} T^{5} + 8067019202447 p^{18} T^{6} - 3010300 p^{27} T^{7} + p^{36} T^{8}$$
29$C_2 \wr S_4$ $$1 - 6153240 T + 66652695791987 T^{2} -$$$$26\!\cdots\!04$$$$T^{3} +$$$$15\!\cdots\!08$$$$T^{4} -$$$$26\!\cdots\!04$$$$p^{9} T^{5} + 66652695791987 p^{18} T^{6} - 6153240 p^{27} T^{7} + p^{36} T^{8}$$
31$C_2 \wr S_4$ $$1 - 12774024 T + 157641860418028 T^{2} -$$$$10\!\cdots\!12$$$$T^{3} +$$$$69\!\cdots\!98$$$$T^{4} -$$$$10\!\cdots\!12$$$$p^{9} T^{5} + 157641860418028 p^{18} T^{6} - 12774024 p^{27} T^{7} + p^{36} T^{8}$$
37$C_2 \wr S_4$ $$1 - 20506048 T + 451640199111340 T^{2} -$$$$57\!\cdots\!32$$$$T^{3} +$$$$77\!\cdots\!30$$$$T^{4} -$$$$57\!\cdots\!32$$$$p^{9} T^{5} + 451640199111340 p^{18} T^{6} - 20506048 p^{27} T^{7} + p^{36} T^{8}$$
41$C_2 \wr S_4$ $$1 - 11620300 T + 777621524436344 T^{2} -$$$$46\!\cdots\!00$$$$T^{3} +$$$$29\!\cdots\!26$$$$T^{4} -$$$$46\!\cdots\!00$$$$p^{9} T^{5} + 777621524436344 p^{18} T^{6} - 11620300 p^{27} T^{7} + p^{36} T^{8}$$
43$C_2 \wr S_4$ $$1 - 7698327 T + 1116374999600644 T^{2} -$$$$20\!\cdots\!79$$$$T^{3} +$$$$59\!\cdots\!74$$$$T^{4} -$$$$20\!\cdots\!79$$$$p^{9} T^{5} + 1116374999600644 p^{18} T^{6} - 7698327 p^{27} T^{7} + p^{36} T^{8}$$
47$C_2 \wr S_4$ $$1 + 31581083 T + 59729788569388 p T^{2} +$$$$86\!\cdots\!23$$$$T^{3} +$$$$43\!\cdots\!46$$$$T^{4} +$$$$86\!\cdots\!23$$$$p^{9} T^{5} + 59729788569388 p^{19} T^{6} + 31581083 p^{27} T^{7} + p^{36} T^{8}$$
53$C_2 \wr S_4$ $$1 - 72549422 T + 9181124351782847 T^{2} -$$$$40\!\cdots\!96$$$$T^{3} +$$$$36\!\cdots\!60$$$$T^{4} -$$$$40\!\cdots\!96$$$$p^{9} T^{5} + 9181124351782847 p^{18} T^{6} - 72549422 p^{27} T^{7} + p^{36} T^{8}$$
59$C_2 \wr S_4$ $$1 + 149234120 T + 38463872258171009 T^{2} +$$$$37\!\cdots\!70$$$$T^{3} +$$$$51\!\cdots\!56$$$$T^{4} +$$$$37\!\cdots\!70$$$$p^{9} T^{5} + 38463872258171009 p^{18} T^{6} + 149234120 p^{27} T^{7} + p^{36} T^{8}$$
61$C_2 \wr S_4$ $$1 - 129004373 T + 36134043355765018 T^{2} -$$$$36\!\cdots\!19$$$$T^{3} +$$$$61\!\cdots\!14$$$$T^{4} -$$$$36\!\cdots\!19$$$$p^{9} T^{5} + 36134043355765018 p^{18} T^{6} - 129004373 p^{27} T^{7} + p^{36} T^{8}$$
67$C_2 \wr S_4$ $$1 - 132595266 T + 50476271812135045 T^{2} -$$$$24\!\cdots\!06$$$$T^{3} +$$$$10\!\cdots\!12$$$$T^{4} -$$$$24\!\cdots\!06$$$$p^{9} T^{5} + 50476271812135045 p^{18} T^{6} - 132595266 p^{27} T^{7} + p^{36} T^{8}$$
71$C_2 \wr S_4$ $$1 + 47138482 T + 109274701906688168 T^{2} -$$$$43\!\cdots\!18$$$$T^{3} +$$$$54\!\cdots\!62$$$$T^{4} -$$$$43\!\cdots\!18$$$$p^{9} T^{5} + 109274701906688168 p^{18} T^{6} + 47138482 p^{27} T^{7} + p^{36} T^{8}$$
73$C_2 \wr S_4$ $$1 + 39332795 T + 147831632963514661 T^{2} -$$$$30\!\cdots\!82$$$$T^{3} +$$$$10\!\cdots\!82$$$$T^{4} -$$$$30\!\cdots\!82$$$$p^{9} T^{5} + 147831632963514661 p^{18} T^{6} + 39332795 p^{27} T^{7} + p^{36} T^{8}$$
79$C_2 \wr S_4$ $$1 + 307010840 T + 250097675403470728 T^{2} +$$$$12\!\cdots\!20$$$$T^{3} +$$$$34\!\cdots\!42$$$$T^{4} +$$$$12\!\cdots\!20$$$$p^{9} T^{5} + 250097675403470728 p^{18} T^{6} + 307010840 p^{27} T^{7} + p^{36} T^{8}$$
83$C_2 \wr S_4$ $$1 + 746568232 T + 900563380389680840 T^{2} +$$$$42\!\cdots\!92$$$$T^{3} +$$$$26\!\cdots\!54$$$$T^{4} +$$$$42\!\cdots\!92$$$$p^{9} T^{5} + 900563380389680840 p^{18} T^{6} + 746568232 p^{27} T^{7} + p^{36} T^{8}$$
89$C_2 \wr S_4$ $$1 - 286943482 T + 949493363525023412 T^{2} -$$$$20\!\cdots\!34$$$$T^{3} +$$$$42\!\cdots\!38$$$$T^{4} -$$$$20\!\cdots\!34$$$$p^{9} T^{5} + 949493363525023412 p^{18} T^{6} - 286943482 p^{27} T^{7} + p^{36} T^{8}$$
97$C_2 \wr S_4$ $$1 - 793519958 T - 696016064440921328 T^{2} -$$$$23\!\cdots\!78$$$$T^{3} +$$$$15\!\cdots\!70$$$$T^{4} -$$$$23\!\cdots\!78$$$$p^{9} T^{5} - 696016064440921328 p^{18} T^{6} - 793519958 p^{27} T^{7} + p^{36} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$