# Properties

 Degree 32 Conductor $3^{16} \cdot 5^{16} \cdot 7^{16}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 32·7-s − 68·16-s + 112·25-s − 384·37-s + 64·43-s + 512·49-s + 320·67-s + 224·81-s − 2.17e3·112-s + 1.13e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 3.58e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
 L(s)  = 1 + 32/7·7-s − 4.25·16-s + 4.47·25-s − 10.3·37-s + 1.48·43-s + 10.4·49-s + 4.77·67-s + 2.76·81-s − 19.4·112-s + 9.38·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 20.4·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$32$$ $$N$$ = $$3^{16} \cdot 5^{16} \cdot 7^{16}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{105} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(32,\ 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )$$ $$L(\frac{3}{2})$$ $$\approx$$ $$6.29083$$ $$L(\frac12)$$ $$\approx$$ $$6.29083$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 32. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad3 $$1 - 224 T^{4} + 290 p^{4} T^{8} - 224 p^{8} T^{12} + p^{16} T^{16}$$
5 $$( 1 - 56 T^{2} + 16 p^{3} T^{4} - 56 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
7 $$( 1 - 16 T + 128 T^{2} - 208 T^{3} - 958 T^{4} - 208 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
good2 $$( 1 + 17 T^{4} + p^{8} T^{8} )^{4}$$
11 $$( 1 - 284 T^{2} + 40742 T^{4} - 284 p^{4} T^{6} + p^{8} T^{8} )^{4}$$
13 $$( 1 - 74400 T^{4} + 2876017858 T^{8} - 74400 p^{8} T^{12} + p^{16} T^{16} )^{2}$$
17 $$( 1 + 280836 T^{4} + 33000880390 T^{8} + 280836 p^{8} T^{12} + p^{16} T^{16} )^{2}$$
19 $$( 1 + 696 T^{2} + 371920 T^{4} + 696 p^{4} T^{6} + p^{8} T^{8} )^{4}$$
23 $$( 1 + 107644 T^{4} - 83134815354 T^{8} + 107644 p^{8} T^{12} + p^{16} T^{16} )^{2}$$
29 $$( 1 + 2444 T^{2} + 2801222 T^{4} + 2444 p^{4} T^{6} + p^{8} T^{8} )^{4}$$
31 $$( 1 - 2596 T^{2} + 3425222 T^{4} - 2596 p^{4} T^{6} + p^{8} T^{8} )^{4}$$
37 $$( 1 + 96 T + 4608 T^{2} + 183264 T^{3} + 6996962 T^{4} + 183264 p^{2} T^{5} + 4608 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{4}$$
41 $$( 1 + 5716 T^{2} + 13775622 T^{4} + 5716 p^{4} T^{6} + p^{8} T^{8} )^{4}$$
43 $$( 1 - 16 T + 128 T^{2} - 2896 T^{3} - 2716702 T^{4} - 2896 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{4}$$
47 $$( 1 - 5511420 T^{4} + 7619082080518 T^{8} - 5511420 p^{8} T^{12} + p^{16} T^{16} )^{2}$$
53 $$( 1 - 19870076 T^{4} + 198820934324166 T^{8} - 19870076 p^{8} T^{12} + p^{16} T^{16} )^{2}$$
59 $$( 1 - 10776 T^{2} + 52783792 T^{4} - 10776 p^{4} T^{6} + p^{8} T^{8} )^{4}$$
61 $$( 1 - 8200 T^{2} + 36103376 T^{4} - 8200 p^{4} T^{6} + p^{8} T^{8} )^{4}$$
67 $$( 1 - 80 T + 3200 T^{2} + 17520 T^{3} - 22069342 T^{4} + 17520 p^{2} T^{5} + 3200 p^{4} T^{6} - 80 p^{6} T^{7} + p^{8} T^{8} )^{4}$$
71 $$( 1 - 17504 T^{2} + 126269762 T^{4} - 17504 p^{4} T^{6} + p^{8} T^{8} )^{4}$$
73 $$( 1 - 10432892 T^{4} + 418480444620678 T^{8} - 10432892 p^{8} T^{12} + p^{16} T^{16} )^{2}$$
79 $$( 1 - 6736 T^{2} + p^{4} T^{4} )^{8}$$
83 $$( 1 - 55419104 T^{4} + 5263624308845250 T^{8} - 55419104 p^{8} T^{12} + p^{16} T^{16} )^{2}$$
89 $$( 1 - 1108 T^{2} - 101931898 T^{4} - 1108 p^{4} T^{6} + p^{8} T^{8} )^{4}$$
97 $$( 1 + 284585220 T^{4} + 35721324461420038 T^{8} + 284585220 p^{8} T^{12} + p^{16} T^{16} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}