# Properties

 Degree 32 Conductor $17^{16} \cdot 43^{16}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 24·13-s + 56·23-s − 64·59-s + 96·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 288·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
 L(s)  = 1 − 6.65·13-s + 11.6·23-s − 8.33·59-s + 10.8·79-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 22.1·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16} \cdot 43^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16} \cdot 43^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$32$$ $$N$$ = $$17^{16} \cdot 43^{16}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{731} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(32,\ 17^{16} \cdot 43^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )$ $L(1)$ $\approx$ $2.09188$ $L(\frac12)$ $\approx$ $2.09188$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{17,\;43\}$,$$F_p(T)$$ is a polynomial of degree 32. If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad17 $$1 + 79967 T^{8} + p^{8} T^{16}$$
43 $$( 1 + p^{4} T^{8} )^{2}$$
good2 $$( 1 + p^{4} T^{8} )^{4}$$
3 $$( 1 + p^{8} T^{16} )^{2}$$
5 $$( 1 + p^{8} T^{16} )^{2}$$
7 $$( 1 + p^{8} T^{16} )^{2}$$
11 $$( 1 + 199 T^{4} + p^{4} T^{8} )^{2}( 1 + 10319 T^{8} + p^{8} T^{16} )$$
13 $$( 1 + 3 T + p T^{2} )^{8}( 1 - 17 T^{2} + p^{2} T^{4} )^{4}$$
19 $$( 1 + p^{4} T^{8} )^{4}$$
23 $$( 1 - 7 T + p T^{2} )^{8}( 1 + 540719 T^{8} + p^{8} T^{16} )$$
29 $$( 1 + p^{8} T^{16} )^{2}$$
31 $$( 1 - 1561 T^{4} + p^{4} T^{8} )^{2}( 1 + 589679 T^{8} + p^{8} T^{16} )$$
37 $$( 1 + p^{8} T^{16} )^{2}$$
41 $$( 1 - 1841 T^{4} + p^{4} T^{8} )^{2}( 1 - 2262241 T^{8} + p^{8} T^{16} )$$
47 $$( 1 - 6983806 T^{8} + p^{8} T^{16} )^{2}$$
53 $$( 1 + 63 T^{2} + p^{2} T^{4} )^{4}( 1 - 1649 T^{4} + p^{4} T^{8} )^{2}$$
59 $$( 1 + 8 T + p T^{2} )^{8}( 1 - 4046 T^{4} + p^{4} T^{8} )^{2}$$
61 $$( 1 + p^{8} T^{16} )^{2}$$
67 $$( 1 - 39816433 T^{8} + p^{8} T^{16} )^{2}$$
71 $$( 1 + p^{8} T^{16} )^{2}$$
73 $$( 1 + p^{8} T^{16} )^{2}$$
79 $$( 1 - 12 T + p T^{2} )^{8}( 1 + 73045634 T^{8} + p^{8} T^{16} )$$
83 $$( 1 - 93091441 T^{8} + p^{8} T^{16} )^{2}$$
89 $$( 1 + p^{2} T^{4} )^{8}$$
97 $$( 1 + 18431 T^{4} + p^{4} T^{8} )^{2}( 1 + 162643199 T^{8} + p^{8} T^{16} )$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}