Properties

 Label 12-95e6-1.1-c1e6-0-1 Degree $12$ Conductor $735091890625$ Sign $1$ Analytic cond. $0.190547$ Root an. cond. $0.870964$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 + 2·4-s − 5-s + 2·9-s + 2·11-s − 3·16-s + 6·19-s − 2·20-s + 2·25-s − 36·29-s + 4·36-s + 12·41-s + 4·44-s − 2·45-s + 23·49-s − 2·55-s − 20·59-s − 14·61-s − 8·64-s + 52·71-s + 12·76-s + 24·79-s + 3·80-s + 81-s − 24·89-s − 6·95-s + 4·99-s + 4·100-s + ⋯
 L(s)  = 1 + 4-s − 0.447·5-s + 2/3·9-s + 0.603·11-s − 3/4·16-s + 1.37·19-s − 0.447·20-s + 2/5·25-s − 6.68·29-s + 2/3·36-s + 1.87·41-s + 0.603·44-s − 0.298·45-s + 23/7·49-s − 0.269·55-s − 2.60·59-s − 1.79·61-s − 64-s + 6.17·71-s + 1.37·76-s + 2.70·79-s + 0.335·80-s + 1/9·81-s − 2.54·89-s − 0.615·95-s + 0.402·99-s + 2/5·100-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$12$$ Conductor: $$5^{6} \cdot 19^{6}$$ Sign: $1$ Analytic conductor: $$0.190547$$ Root analytic conductor: $$0.870964$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{95} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.030895812$$ $$L(\frac12)$$ $$\approx$$ $$1.030895812$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + T - T^{2} - 2 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
19 $$( 1 - T )^{6}$$
good2 $$1 - p T^{2} + 7 T^{4} - 3 p^{2} T^{6} + 7 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12}$$
3 $$1 - 2 T^{2} + p T^{4} - 20 T^{6} + p^{3} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12}$$
7 $$1 - 23 T^{2} + 307 T^{4} - 2586 T^{6} + 307 p^{2} T^{8} - 23 p^{4} T^{10} + p^{6} T^{12}$$
11 $$( 1 - T + 17 T^{2} - 10 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2}$$
13 $$1 - 50 T^{2} + 1315 T^{4} - 21108 T^{6} + 1315 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12}$$
17 $$( 1 - 9 T + 19 T^{2} + 18 T^{3} + 19 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )( 1 + 9 T + 19 T^{2} - 18 T^{3} + 19 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )$$
23 $$1 - 102 T^{2} + 4831 T^{4} - 138580 T^{6} + 4831 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12}$$
29 $$( 1 + 6 T + p T^{2} )^{6}$$
31 $$( 1 + 37 T^{2} + 128 T^{3} + 37 p T^{4} + p^{3} T^{6} )^{2}$$
37 $$1 - 166 T^{2} + 13011 T^{4} - 608316 T^{6} + 13011 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12}$$
41 $$( 1 - 6 T + 79 T^{2} - 516 T^{3} + 79 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
43 $$1 - 239 T^{2} + 24571 T^{4} - 1388154 T^{6} + 24571 p^{2} T^{8} - 239 p^{4} T^{10} + p^{6} T^{12}$$
47 $$1 - 95 T^{2} + 5443 T^{4} - 214314 T^{6} + 5443 p^{2} T^{8} - 95 p^{4} T^{10} + p^{6} T^{12}$$
53 $$1 - 162 T^{2} + 11539 T^{4} - 610708 T^{6} + 11539 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12}$$
59 $$( 1 + 10 T + 185 T^{2} + 1132 T^{3} + 185 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
61 $$( 1 + 7 T + 79 T^{2} + 78 T^{3} + 79 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
67 $$1 - 62 T^{2} + 4771 T^{4} - 199788 T^{6} + 4771 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12}$$
71 $$( 1 - 26 T + 413 T^{2} - 4124 T^{3} + 413 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
73 $$1 - 307 T^{2} + 43299 T^{4} - 3822498 T^{6} + 43299 p^{2} T^{8} - 307 p^{4} T^{10} + p^{6} T^{12}$$
79 $$( 1 - 12 T + 229 T^{2} - 1864 T^{3} + 229 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
83 $$1 - 270 T^{2} + 39367 T^{4} - 3817060 T^{6} + 39367 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12}$$
89 $$( 1 + 12 T - 17 T^{2} - 1320 T^{3} - 17 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
97 $$1 - 554 T^{2} + 130507 T^{4} - 16717956 T^{6} + 130507 p^{2} T^{8} - 554 p^{4} T^{10} + p^{6} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$