Properties

 Label 8-980e4-1.1-c1e4-0-14 Degree $8$ Conductor $922368160000$ Sign $1$ Analytic cond. $3749.83$ Root an. cond. $2.79738$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 6·3-s + 2·5-s + 15·9-s + 3·11-s − 12·15-s + 9·17-s + 19-s + 12·23-s − 7·25-s − 18·27-s − 2·29-s − 31-s − 18·33-s − 27·37-s + 30·41-s + 30·45-s − 15·47-s − 54·51-s + 3·53-s + 6·55-s − 6·57-s − 59-s − 12·61-s + 18·67-s − 72·69-s + 12·71-s − 15·73-s + ⋯
 L(s)  = 1 − 3.46·3-s + 0.894·5-s + 5·9-s + 0.904·11-s − 3.09·15-s + 2.18·17-s + 0.229·19-s + 2.50·23-s − 7/5·25-s − 3.46·27-s − 0.371·29-s − 0.179·31-s − 3.13·33-s − 4.43·37-s + 4.68·41-s + 4.47·45-s − 2.18·47-s − 7.56·51-s + 0.412·53-s + 0.809·55-s − 0.794·57-s − 0.130·59-s − 1.53·61-s + 2.19·67-s − 8.66·69-s + 1.42·71-s − 1.75·73-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$2^{8} \cdot 5^{4} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$3749.83$$ Root analytic conductor: $$2.79738$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{980} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.9290553517$$ $$L(\frac12)$$ $$\approx$$ $$0.9290553517$$ $$L(\frac{3}{2})$$ 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 $$1$$
5$C_2$ $$( 1 - T + p T^{2} )^{2}$$
7 $$1$$
good3$C_2$ $$( 1 + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2}$$
11$D_4\times C_2$ $$1 - 3 T - T^{2} + 36 T^{3} - 120 T^{4} + 36 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
13$D_4\times C_2$ $$1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}$$
17$D_4\times C_2$ $$1 - 9 T + 63 T^{2} - 324 T^{3} + 1466 T^{4} - 324 p T^{5} + 63 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
19$D_4\times C_2$ $$1 - T - 23 T^{2} + 14 T^{3} + 196 T^{4} + 14 p T^{5} - 23 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
23$C_2$$\times$$C_2^2$ $$( 1 - 4 T + p T^{2} )^{2}( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )$$
29$D_{4}$ $$( 1 + T + 44 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 + T - 47 T^{2} - 14 T^{3} + 1312 T^{4} - 14 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 + 27 T + 373 T^{2} + 3510 T^{3} + 24522 T^{4} + 3510 p T^{5} + 373 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 - 15 T + 124 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 125 T^{2} + 7248 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 + 15 T + 183 T^{2} + 1620 T^{3} + 12980 T^{4} + 1620 p T^{5} + 183 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 3 T + 67 T^{2} - 192 T^{3} + 1446 T^{4} - 192 p T^{5} + 67 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 + T - 103 T^{2} - 14 T^{3} + 7276 T^{4} - 14 p T^{5} - 103 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 + 12 T + 43 T^{2} - 252 T^{3} - 2304 T^{4} - 252 p T^{5} + 43 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 18 T + 193 T^{2} - 1530 T^{3} + 9972 T^{4} - 1530 p T^{5} + 193 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 15 T + 235 T^{2} + 2400 T^{3} + 25746 T^{4} + 2400 p T^{5} + 235 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 - 7 T - 107 T^{2} + 14 T^{3} + 14224 T^{4} + 14 p T^{5} - 107 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
83$D_4\times C_2$ $$1 - 245 T^{2} + 28656 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8}$$
89$C_2^2$ $$( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 - 146 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$