# Properties

 Degree $6$ Conductor $15625$ Sign $-1$ Motivic weight $9$ Primitive no Self-dual yes Analytic rank $3$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 33·2-s − 89·3-s − 53·4-s + 2.93e3·6-s − 5.25e3·7-s + 1.93e4·8-s + 3.55e3·9-s − 5.46e4·11-s + 4.71e3·12-s − 2.15e5·13-s + 1.73e5·14-s − 5.02e5·16-s − 3.34e5·17-s − 1.17e5·18-s + 8.18e5·19-s + 4.67e5·21-s + 1.80e6·22-s − 3.52e6·23-s − 1.72e6·24-s + 7.12e6·26-s − 1.70e6·27-s + 2.78e5·28-s + 2.17e6·29-s + 4.27e6·31-s + 6.15e6·32-s + 4.86e6·33-s + 1.10e7·34-s + ⋯
 L(s)  = 1 − 1.45·2-s − 0.634·3-s − 0.103·4-s + 0.925·6-s − 0.827·7-s + 1.67·8-s + 0.180·9-s − 1.12·11-s + 0.0656·12-s − 2.09·13-s + 1.20·14-s − 1.91·16-s − 0.972·17-s − 0.263·18-s + 1.44·19-s + 0.525·21-s + 1.64·22-s − 2.62·23-s − 1.06·24-s + 3.05·26-s − 0.618·27-s + 0.0856·28-s + 0.571·29-s + 0.831·31-s + 1.03·32-s + 0.714·33-s + 1.41·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$15625$$    =    $$5^{6}$$ Sign: $-1$ Motivic weight: $$9$$ Character: induced by $\chi_{25} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 15625,\ (\ :9/2, 9/2, 9/2),\ -1)$$

## Particular Values

 $$L(5)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{11}{2})$$ 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)$
bad5 $$1$$
good2$S_4\times C_2$ $$1 + 33 T + 571 p T^{2} + 2505 p^{3} T^{3} + 571 p^{10} T^{4} + 33 p^{18} T^{5} + p^{27} T^{6}$$
3$S_4\times C_2$ $$1 + 89 T + 1456 p T^{2} + 197885 p^{2} T^{3} + 1456 p^{10} T^{4} + 89 p^{18} T^{5} + p^{27} T^{6}$$
7$S_4\times C_2$ $$1 + 5258 T + 11082951 p T^{2} + 9176976100 p^{2} T^{3} + 11082951 p^{10} T^{4} + 5258 p^{18} T^{5} + p^{27} T^{6}$$
11$S_4\times C_2$ $$1 + 54699 T + 5816187440 T^{2} + 182671409832855 T^{3} + 5816187440 p^{9} T^{4} + 54699 p^{18} T^{5} + p^{27} T^{6}$$
13$S_4\times C_2$ $$1 + 215884 T + 31225156343 T^{2} + 3094983572287480 T^{3} + 31225156343 p^{9} T^{4} + 215884 p^{18} T^{5} + p^{27} T^{6}$$
17$S_4\times C_2$ $$1 + 334983 T + 355030662062 T^{2} + 77675997792847395 T^{3} + 355030662062 p^{9} T^{4} + 334983 p^{18} T^{5} + p^{27} T^{6}$$
19$S_4\times C_2$ $$1 - 818845 T + 468173656712 T^{2} - 302339390932836385 T^{3} + 468173656712 p^{9} T^{4} - 818845 p^{18} T^{5} + p^{27} T^{6}$$
23$S_4\times C_2$ $$1 + 3526854 T + 8700596724993 T^{2} + 13086552252942250620 T^{3} + 8700596724993 p^{9} T^{4} + 3526854 p^{18} T^{5} + p^{27} T^{6}$$
29$S_4\times C_2$ $$1 - 2175480 T + 1119231270883 p T^{2} - 59155485309271560240 T^{3} + 1119231270883 p^{10} T^{4} - 2175480 p^{18} T^{5} + p^{27} T^{6}$$
31$S_4\times C_2$ $$1 - 4274066 T + 82851493809465 T^{2} -$$$$22\!\cdots\!20$$$$T^{3} + 82851493809465 p^{9} T^{4} - 4274066 p^{18} T^{5} + p^{27} T^{6}$$
37$S_4\times C_2$ $$1 - 10305042 T + 340842196290747 T^{2} -$$$$25\!\cdots\!40$$$$T^{3} + 340842196290747 p^{9} T^{4} - 10305042 p^{18} T^{5} + p^{27} T^{6}$$
41$S_4\times C_2$ $$1 - 5926311 T + 232043727124790 T^{2} -$$$$23\!\cdots\!95$$$$T^{3} + 232043727124790 p^{9} T^{4} - 5926311 p^{18} T^{5} + p^{27} T^{6}$$
43$S_4\times C_2$ $$1 - 24429956 T + 1676050268074193 T^{2} -$$$$57\!\cdots\!00$$$$p T^{3} + 1676050268074193 p^{9} T^{4} - 24429956 p^{18} T^{5} + p^{27} T^{6}$$
47$S_4\times C_2$ $$1 + 66858708 T + 4517266130065277 T^{2} +$$$$15\!\cdots\!80$$$$T^{3} + 4517266130065277 p^{9} T^{4} + 66858708 p^{18} T^{5} + p^{27} T^{6}$$
53$S_4\times C_2$ $$1 + 132620514 T + 9209248487675243 T^{2} +$$$$47\!\cdots\!40$$$$T^{3} + 9209248487675243 p^{9} T^{4} + 132620514 p^{18} T^{5} + p^{27} T^{6}$$
59$S_4\times C_2$ $$1 - 5670960 T + 19337695182532817 T^{2} -$$$$14\!\cdots\!80$$$$T^{3} + 19337695182532817 p^{9} T^{4} - 5670960 p^{18} T^{5} + p^{27} T^{6}$$
61$S_4\times C_2$ $$1 - 125306926 T + 22192130203358915 T^{2} -$$$$23\!\cdots\!20$$$$T^{3} + 22192130203358915 p^{9} T^{4} - 125306926 p^{18} T^{5} + p^{27} T^{6}$$
67$S_4\times C_2$ $$1 + 88829483 T + 30562384919894712 T^{2} +$$$$76\!\cdots\!95$$$$T^{3} + 30562384919894712 p^{9} T^{4} + 88829483 p^{18} T^{5} + p^{27} T^{6}$$
71$S_4\times C_2$ $$1 - 297550596 T + 87036096018332165 T^{2} -$$$$15\!\cdots\!20$$$$T^{3} + 87036096018332165 p^{9} T^{4} - 297550596 p^{18} T^{5} + p^{27} T^{6}$$
73$S_4\times C_2$ $$1 + 181321729 T + 94365252526618118 T^{2} +$$$$75\!\cdots\!45$$$$T^{3} + 94365252526618118 p^{9} T^{4} + 181321729 p^{18} T^{5} + p^{27} T^{6}$$
79$S_4\times C_2$ $$1 + 310025170 T + 176092553119892457 T^{2} +$$$$48\!\cdots\!60$$$$T^{3} + 176092553119892457 p^{9} T^{4} + 310025170 p^{18} T^{5} + p^{27} T^{6}$$
83$S_4\times C_2$ $$1 - 731088801 T + 685433348047337568 T^{2} -$$$$27\!\cdots\!65$$$$T^{3} + 685433348047337568 p^{9} T^{4} - 731088801 p^{18} T^{5} + p^{27} T^{6}$$
89$S_4\times C_2$ $$1 + 1103860035 T + 1320664213408319502 T^{2} +$$$$75\!\cdots\!55$$$$T^{3} + 1320664213408319502 p^{9} T^{4} + 1103860035 p^{18} T^{5} + p^{27} T^{6}$$
97$S_4\times C_2$ $$1 - 332236842 T + 1487745026246658927 T^{2} -$$$$16\!\cdots\!20$$$$T^{3} + 1487745026246658927 p^{9} T^{4} - 332236842 p^{18} T^{5} + p^{27} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$