# Properties

 Label 8-15e4-1.1-c3e4-0-0 Degree $8$ Conductor $50625$ Sign $1$ Analytic cond. $0.613520$ Root an. cond. $0.940759$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 7·4-s + 6·5-s − 18·9-s − 84·11-s + 16-s − 112·19-s + 42·20-s + 146·25-s + 636·29-s + 104·31-s − 126·36-s − 816·41-s − 588·44-s − 108·45-s + 616·49-s − 504·55-s + 372·59-s + 680·61-s + 119·64-s − 72·71-s − 784·76-s − 760·79-s + 6·80-s + 243·81-s − 2.23e3·89-s − 672·95-s + 1.51e3·99-s + ⋯
 L(s)  = 1 + 7/8·4-s + 0.536·5-s − 2/3·9-s − 2.30·11-s + 1/64·16-s − 1.35·19-s + 0.469·20-s + 1.16·25-s + 4.07·29-s + 0.602·31-s − 0.583·36-s − 3.10·41-s − 2.01·44-s − 0.357·45-s + 1.79·49-s − 1.23·55-s + 0.820·59-s + 1.42·61-s + 0.232·64-s − 0.120·71-s − 1.18·76-s − 1.08·79-s + 0.00838·80-s + 1/3·81-s − 2.65·89-s − 0.725·95-s + 1.53·99-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$50625$$    =    $$3^{4} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$0.613520$$ Root analytic conductor: $$0.940759$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 50625,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.045618377$$ $$L(\frac12)$$ $$\approx$$ $$1.045618377$$ $$L(\frac{5}{2})$$ 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)$
bad3$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
5$C_2^2$ $$1 - 6 T - 22 p T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}$$
good2$D_4\times C_2$ $$1 - 7 T^{2} + 3 p^{4} T^{4} - 7 p^{6} T^{6} + p^{12} T^{8}$$
7$D_4\times C_2$ $$1 - 88 p T^{2} + 316878 T^{4} - 88 p^{7} T^{6} + p^{12} T^{8}$$
11$D_{4}$ $$( 1 + 42 T + 2734 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 5008 T^{2} + 13678638 T^{4} - 5008 p^{6} T^{6} + p^{12} T^{8}$$
17$D_4\times C_2$ $$1 - 12400 T^{2} + 76051038 T^{4} - 12400 p^{6} T^{6} + p^{12} T^{8}$$
19$D_{4}$ $$( 1 + 56 T + 8598 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 46204 T^{2} + 828262758 T^{4} - 46204 p^{6} T^{6} + p^{12} T^{8}$$
29$D_{4}$ $$( 1 - 318 T + 55978 T^{2} - 318 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
31$D_{4}$ $$( 1 - 52 T + 58782 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 96016 T^{2} + 4637534478 T^{4} - 96016 p^{6} T^{6} + p^{12} T^{8}$$
41$D_{4}$ $$( 1 + 408 T + 177982 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 121900 T^{2} + 14997346998 T^{4} - 121900 p^{6} T^{6} + p^{12} T^{8}$$
47$D_4\times C_2$ $$1 - 225580 T^{2} + 31469120358 T^{4} - 225580 p^{6} T^{6} + p^{12} T^{8}$$
53$D_4\times C_2$ $$1 - 376864 T^{2} + 68697431598 T^{4} - 376864 p^{6} T^{6} + p^{12} T^{8}$$
59$D_{4}$ $$( 1 - 186 T + 419038 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
61$D_{4}$ $$( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 861340 T^{2} + 346621507638 T^{4} - 861340 p^{6} T^{6} + p^{12} T^{8}$$
71$D_{4}$ $$( 1 + 36 T + 384046 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 429844 T^{2} + 136740794118 T^{4} - 429844 p^{6} T^{6} + p^{12} T^{8}$$
79$D_{4}$ $$( 1 + 380 T + 99678 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 916108 T^{2} + 434315280918 T^{4} - 916108 p^{6} T^{6} + p^{12} T^{8}$$
89$D_{4}$ $$( 1 + 1116 T + 1508758 T^{2} + 1116 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 - 2174980 T^{2} + 2500346420358 T^{4} - 2174980 p^{6} T^{6} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$