# Properties

 Degree $6$ Conductor $64$ Sign $1$ Motivic weight $33$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 9.24e7·3-s − 5.38e10·5-s + 4.54e12·7-s − 1.04e15·9-s + 2.27e17·11-s + 2.72e17·13-s − 4.98e18·15-s + 9.30e19·17-s + 1.32e21·19-s + 4.20e20·21-s + 1.16e22·23-s − 4.40e22·25-s − 1.43e23·27-s + 2.89e24·29-s + 1.58e25·31-s + 2.10e25·33-s − 2.44e23·35-s + 3.47e25·37-s + 2.52e25·39-s − 6.73e26·41-s − 1.55e27·43-s + 5.63e25·45-s − 7.22e27·47-s − 1.39e28·49-s + 8.60e27·51-s + 5.40e28·53-s − 1.22e28·55-s + ⋯
 L(s)  = 1 + 1.24·3-s − 0.157·5-s + 0.0516·7-s − 0.187·9-s + 1.49·11-s + 0.113·13-s − 0.195·15-s + 0.463·17-s + 1.05·19-s + 0.0640·21-s + 0.397·23-s − 0.378·25-s − 0.345·27-s + 2.15·29-s + 3.90·31-s + 1.85·33-s − 0.00815·35-s + 0.463·37-s + 0.141·39-s − 1.65·41-s − 1.73·43-s + 0.0296·45-s − 1.85·47-s − 1.80·49-s + 0.575·51-s + 1.91·53-s − 0.235·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+33/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$64$$    =    $$2^{6}$$ Sign: $1$ Motivic weight: $$33$$ Character: induced by $\chi_{4} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 64,\ (\ :33/2, 33/2, 33/2),\ 1)$$

## Particular Values

 $$L(17)$$ $$\approx$$ $$7.111810932$$ $$L(\frac12)$$ $$\approx$$ $$7.111810932$$ $$L(\frac{35}{2})$$ 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 $$1$$
good3$S_4\times C_2$ $$1 - 30830596 p T + 13168414343905 p^{6} T^{2} - 527785882937478232 p^{13} T^{3} + 13168414343905 p^{39} T^{4} - 30830596 p^{67} T^{5} + p^{99} T^{6}$$
5$S_4\times C_2$ $$1 + 53880683886 T +$$$$37\!\cdots\!83$$$$p^{3} T^{2} -$$$$11\!\cdots\!32$$$$p^{8} T^{3} +$$$$37\!\cdots\!83$$$$p^{36} T^{4} + 53880683886 p^{66} T^{5} + p^{99} T^{6}$$
7$S_4\times C_2$ $$1 - 648715702056 p T +$$$$28\!\cdots\!53$$$$p^{2} T^{2} -$$$$33\!\cdots\!20$$$$p^{6} T^{3} +$$$$28\!\cdots\!53$$$$p^{35} T^{4} - 648715702056 p^{67} T^{5} + p^{99} T^{6}$$
11$S_4\times C_2$ $$1 - 20692514300222700 p T +$$$$16\!\cdots\!33$$$$p^{2} T^{2} +$$$$53\!\cdots\!00$$$$p^{3} T^{3} +$$$$16\!\cdots\!33$$$$p^{35} T^{4} - 20692514300222700 p^{67} T^{5} + p^{99} T^{6}$$
13$S_4\times C_2$ $$1 - 272970442217358762 T +$$$$10\!\cdots\!15$$$$p T^{2} -$$$$12\!\cdots\!32$$$$p^{3} T^{3} +$$$$10\!\cdots\!15$$$$p^{34} T^{4} - 272970442217358762 p^{66} T^{5} + p^{99} T^{6}$$
17$S_4\times C_2$ $$1 - 93037188311816716854 T +$$$$10\!\cdots\!75$$$$T^{2} -$$$$44\!\cdots\!56$$$$p T^{3} +$$$$10\!\cdots\!75$$$$p^{33} T^{4} - 93037188311816716854 p^{66} T^{5} + p^{99} T^{6}$$
19$S_4\times C_2$ $$1 - 3670049324584713948 p^{2} T +$$$$20\!\cdots\!95$$$$p T^{2} -$$$$97\!\cdots\!80$$$$p^{2} T^{3} +$$$$20\!\cdots\!95$$$$p^{34} T^{4} - 3670049324584713948 p^{68} T^{5} + p^{99} T^{6}$$
23$S_4\times C_2$ $$1 -$$$$11\!\cdots\!96$$$$T +$$$$81\!\cdots\!51$$$$p T^{2} -$$$$28\!\cdots\!60$$$$p^{2} T^{3} +$$$$81\!\cdots\!51$$$$p^{34} T^{4} -$$$$11\!\cdots\!96$$$$p^{66} T^{5} + p^{99} T^{6}$$
29$S_4\times C_2$ $$1 -$$$$28\!\cdots\!78$$$$T +$$$$15\!\cdots\!55$$$$p T^{2} -$$$$56\!\cdots\!60$$$$p^{2} T^{3} +$$$$15\!\cdots\!55$$$$p^{34} T^{4} -$$$$28\!\cdots\!78$$$$p^{66} T^{5} + p^{99} T^{6}$$
31$S_4\times C_2$ $$1 -$$$$51\!\cdots\!88$$$$p T +$$$$13\!\cdots\!41$$$$p^{2} T^{2} -$$$$22\!\cdots\!92$$$$p^{3} T^{3} +$$$$13\!\cdots\!41$$$$p^{35} T^{4} -$$$$51\!\cdots\!88$$$$p^{67} T^{5} + p^{99} T^{6}$$
37$S_4\times C_2$ $$1 -$$$$34\!\cdots\!42$$$$T +$$$$58\!\cdots\!87$$$$T^{2} -$$$$50\!\cdots\!80$$$$T^{3} +$$$$58\!\cdots\!87$$$$p^{33} T^{4} -$$$$34\!\cdots\!42$$$$p^{66} T^{5} + p^{99} T^{6}$$
41$S_4\times C_2$ $$1 +$$$$67\!\cdots\!62$$$$T +$$$$59\!\cdots\!11$$$$T^{2} +$$$$21\!\cdots\!68$$$$T^{3} +$$$$59\!\cdots\!11$$$$p^{33} T^{4} +$$$$67\!\cdots\!62$$$$p^{66} T^{5} + p^{99} T^{6}$$
43$S_4\times C_2$ $$1 +$$$$15\!\cdots\!20$$$$T +$$$$22\!\cdots\!29$$$$T^{2} +$$$$45\!\cdots\!40$$$$p T^{3} +$$$$22\!\cdots\!29$$$$p^{33} T^{4} +$$$$15\!\cdots\!20$$$$p^{66} T^{5} + p^{99} T^{6}$$
47$S_4\times C_2$ $$1 +$$$$72\!\cdots\!76$$$$T +$$$$60\!\cdots\!05$$$$T^{2} +$$$$22\!\cdots\!28$$$$T^{3} +$$$$60\!\cdots\!05$$$$p^{33} T^{4} +$$$$72\!\cdots\!76$$$$p^{66} T^{5} + p^{99} T^{6}$$
53$S_4\times C_2$ $$1 -$$$$54\!\cdots\!34$$$$T +$$$$28\!\cdots\!83$$$$T^{2} -$$$$80\!\cdots\!40$$$$T^{3} +$$$$28\!\cdots\!83$$$$p^{33} T^{4} -$$$$54\!\cdots\!34$$$$p^{66} T^{5} + p^{99} T^{6}$$
59$S_4\times C_2$ $$1 -$$$$20\!\cdots\!76$$$$T +$$$$56\!\cdots\!29$$$$T^{2} -$$$$65\!\cdots\!96$$$$T^{3} +$$$$56\!\cdots\!29$$$$p^{33} T^{4} -$$$$20\!\cdots\!76$$$$p^{66} T^{5} + p^{99} T^{6}$$
61$S_4\times C_2$ $$1 -$$$$59\!\cdots\!66$$$$T +$$$$26\!\cdots\!95$$$$T^{2} -$$$$28\!\cdots\!40$$$$T^{3} +$$$$26\!\cdots\!95$$$$p^{33} T^{4} -$$$$59\!\cdots\!66$$$$p^{66} T^{5} + p^{99} T^{6}$$
67$S_4\times C_2$ $$1 +$$$$29\!\cdots\!28$$$$T +$$$$81\!\cdots\!37$$$$T^{2} +$$$$11\!\cdots\!00$$$$T^{3} +$$$$81\!\cdots\!37$$$$p^{33} T^{4} +$$$$29\!\cdots\!28$$$$p^{66} T^{5} + p^{99} T^{6}$$
71$S_4\times C_2$ $$1 -$$$$16\!\cdots\!44$$$$T +$$$$30\!\cdots\!45$$$$T^{2} -$$$$32\!\cdots\!60$$$$T^{3} +$$$$30\!\cdots\!45$$$$p^{33} T^{4} -$$$$16\!\cdots\!44$$$$p^{66} T^{5} + p^{99} T^{6}$$
73$S_4\times C_2$ $$1 -$$$$13\!\cdots\!02$$$$T +$$$$36\!\cdots\!75$$$$T^{2} -$$$$19\!\cdots\!84$$$$T^{3} +$$$$36\!\cdots\!75$$$$p^{33} T^{4} -$$$$13\!\cdots\!02$$$$p^{66} T^{5} + p^{99} T^{6}$$
79$S_4\times C_2$ $$1 -$$$$43\!\cdots\!04$$$$T +$$$$15\!\cdots\!89$$$$T^{2} -$$$$35\!\cdots\!44$$$$T^{3} +$$$$15\!\cdots\!89$$$$p^{33} T^{4} -$$$$43\!\cdots\!04$$$$p^{66} T^{5} + p^{99} T^{6}$$
83$S_4\times C_2$ $$1 +$$$$71\!\cdots\!64$$$$T +$$$$66\!\cdots\!93$$$$T^{2} +$$$$26\!\cdots\!20$$$$T^{3} +$$$$66\!\cdots\!93$$$$p^{33} T^{4} +$$$$71\!\cdots\!64$$$$p^{66} T^{5} + p^{99} T^{6}$$
89$S_4\times C_2$ $$1 +$$$$19\!\cdots\!98$$$$T +$$$$63\!\cdots\!75$$$$T^{2} +$$$$76\!\cdots\!20$$$$T^{3} +$$$$63\!\cdots\!75$$$$p^{33} T^{4} +$$$$19\!\cdots\!98$$$$p^{66} T^{5} + p^{99} T^{6}$$
97$S_4\times C_2$ $$1 -$$$$56\!\cdots\!22$$$$T -$$$$50\!\cdots\!73$$$$T^{2} +$$$$35\!\cdots\!00$$$$T^{3} -$$$$50\!\cdots\!73$$$$p^{33} T^{4} -$$$$56\!\cdots\!22$$$$p^{66} T^{5} + p^{99} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$