# Properties

 Degree 24 Conductor $2^{36} \cdot 3^{12} \cdot 5^{24}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4-s − 6·9-s + 16-s − 32·31-s + 6·36-s − 8·41-s − 36·49-s + 7·64-s + 32·71-s − 32·79-s + 21·81-s + 40·89-s + 68·121-s + 32·124-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + 149-s + 151-s + 157-s + 163-s + 8·164-s + 167-s + 60·169-s + 173-s + ⋯
 L(s)  = 1 − 1/2·4-s − 2·9-s + 1/4·16-s − 5.74·31-s + 36-s − 1.24·41-s − 5.14·49-s + 7/8·64-s + 3.79·71-s − 3.60·79-s + 7/3·81-s + 4.23·89-s + 6.18·121-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.624·164-s + 0.0773·167-s + 4.61·169-s + 0.0760·173-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$24$$ $$N$$ = $$2^{36} \cdot 3^{12} \cdot 5^{24}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{600} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(24,\ 2^{36} \cdot 3^{12} \cdot 5^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )$$ $$L(1)$$ $$\approx$$ $$1.65779$$ $$L(\frac12)$$ $$\approx$$ $$1.65779$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 24. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad2 $$1 + T^{2} - p^{3} T^{6} + p^{4} T^{10} + p^{6} T^{12}$$
3 $$( 1 + T^{2} )^{6}$$
5 $$1$$
good7 $$( 1 + 18 T^{2} + 191 T^{4} + 1532 T^{6} + 191 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
11 $$( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 503 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
13 $$( 1 - 30 T^{2} + 743 T^{4} - 10436 T^{6} + 743 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
17 $$( 1 + 66 T^{2} + 2255 T^{4} + 47324 T^{6} + 2255 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
19 $$( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 1367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
23 $$( 1 + 2 p T^{2} + 1775 T^{4} + 40932 T^{6} + 1775 p^{2} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} )^{2}$$
29 $$( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 3207 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
31 $$( 1 + 8 T + 89 T^{2} + 432 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
37 $$( 1 - 158 T^{2} + 11191 T^{4} - 496772 T^{6} + 11191 p^{2} T^{8} - 158 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
41 $$( 1 + 2 T + 23 T^{2} + 220 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
43 $$( 1 - 130 T^{2} + 9815 T^{4} - 518268 T^{6} + 9815 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
47 $$( 1 + 222 T^{2} + 22367 T^{4} + 1328324 T^{6} + 22367 p^{2} T^{8} + 222 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
53 $$( 1 - 238 T^{2} + 26391 T^{4} - 1758052 T^{6} + 26391 p^{2} T^{8} - 238 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
59 $$( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 20567 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
61 $$( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 20039 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
67 $$( 1 - 274 T^{2} + 37127 T^{4} - 3112476 T^{6} + 37127 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
71 $$( 1 - 8 T + 133 T^{2} - 1008 T^{3} + 133 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
73 $$( 1 + 54 T^{2} + 2367 T^{4} + 531700 T^{6} + 2367 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
79 $$( 1 + 8 T + 233 T^{2} + 1200 T^{3} + 233 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
83 $$( 1 - 306 T^{2} + 50855 T^{4} - 5168732 T^{6} + 50855 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
89 $$( 1 - 10 T + 103 T^{2} - 396 T^{3} + 103 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
97 $$( 1 + 246 T^{2} + 39183 T^{4} + 4535476 T^{6} + 39183 p^{2} T^{8} + 246 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}