# Properties

 Degree 16 Conductor $2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 40·25-s + 56·49-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-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 + 241-s + ⋯
 L(s)  = 1 − 8·25-s + 8·49-s − 4·121-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 + 1.07·169-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 + 0.0644·241-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{7524} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(16,\ 2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$ $L(1)$ $\approx$ $2.945096751$ $L(\frac12)$ $\approx$ $2.945096751$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;11,\;19\}$, $$F_p$$ is a polynomial of degree 16. If $p \in \{2,\;3,\;11,\;19\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad2 $$1$$
3 $$1$$
11 $$( 1 + p T^{2} )^{4}$$
19 $$( 1 - p T^{2} )^{4}$$
good5 $$( 1 + p T^{2} )^{8}$$
7 $$( 1 - p T^{2} )^{8}$$
13 $$( 1 - 7 T^{2} - 120 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
17 $$( 1 + 23 T^{2} + 240 T^{4} + 23 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
23 $$( 1 + p T^{2} )^{8}$$
29 $$( 1 + p T^{2} )^{8}$$
31 $$( 1 - p T^{2} )^{8}$$
37 $$( 1 - p T^{2} )^{8}$$
41 $$( 1 + p T^{2} )^{8}$$
43 $$( 1 - p T^{2} )^{8}$$
47 $$( 1 + p T^{2} )^{8}$$
53 $$( 1 - 103 T^{2} + 7800 T^{4} - 103 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
59 $$( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
61 $$( 1 - p T^{2} )^{8}$$
67 $$( 1 - p T^{2} )^{8}$$
71 $$( 1 - 67 T^{2} - 552 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
73 $$( 1 - p T^{2} )^{8}$$
79 $$( 1 - 139 T^{2} + 13080 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
83 $$( 1 - 109 T^{2} + 4992 T^{4} - 109 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
89 $$( 1 - 31 T^{2} - 6960 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
97 $$( 1 - p T^{2} )^{8}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}