# Properties

 Degree 32 Conductor $2^{96} \cdot 5^{16}$ Sign $1$ Motivic weight 5 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 992·9-s + 440·25-s − 1.00e5·41-s + 1.98e5·49-s + 3.37e5·81-s − 4.59e5·89-s + 2.30e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.04e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 4.36e5·225-s + ⋯
 L(s)  = 1 − 4.08·9-s + 0.140·25-s − 9.32·41-s + 11.8·49-s + 5.71·81-s − 6.14·89-s + 14.3·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 10.9·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s − 0.574·225-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$32$$ $$N$$ = $$2^{96} \cdot 5^{16}$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : induced by $\chi_{320} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(32,\ 2^{96} \cdot 5^{16} ,\ ( \ : [5/2]^{16} ),\ 1 )$$ $$L(3)$$ $$\approx$$ $$2.842403078$$ $$L(\frac12)$$ $$\approx$$ $$2.842403078$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 32. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 $$1$$
5 $$( 1 - 44 p T^{2} - 48914 p^{3} T^{4} - 44 p^{11} T^{6} + p^{20} T^{8} )^{2}$$
good3 $$( 1 + 248 T^{2} + 23110 p T^{4} + 248 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
7 $$( 1 - 49648 T^{2} + 1179577874 T^{4} - 49648 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
11 $$( 1 - 577284 T^{2} + 134071408310 T^{4} - 577284 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
13 $$( 1 + 1011892 T^{2} + 531287817014 T^{4} + 1011892 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
17 $$( 1 + 554828 T^{2} + 3105096833510 T^{4} + 554828 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
19 $$( 1 - 8279076 T^{2} + 29313371734870 T^{4} - 8279076 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
23 $$( 1 - 265392 T^{2} + 72532989580114 T^{4} - 265392 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
29 $$( 1 - 18262996 T^{2} + 467877918484406 T^{4} - 18262996 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
31 $$( 1 + 39407004 T^{2} + 689994929185606 T^{4} + 39407004 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
37 $$( 1 + 171738148 T^{2} + 16366808202619574 T^{4} + 171738148 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
41 $$( 1 + 12544 T + 129356290 T^{2} + 12544 p^{5} T^{3} + p^{10} T^{4} )^{8}$$
43 $$( 1 + 515310072 T^{2} + 108506205300479794 T^{4} + 515310072 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
47 $$( 1 - 112031408 T^{2} + 100831510268064114 T^{4} - 112031408 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
53 $$( 1 + 823212052 T^{2} + 458582260389137174 T^{4} + 823212052 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
59 $$( 1 - 2409567364 T^{2} + 2462718757251169526 T^{4} - 2409567364 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
61 $$( 1 - 2000262204 T^{2} + 2103804668157499606 T^{4} - 2000262204 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
67 $$( 1 + 3461519512 T^{2} + 6000732802449327570 T^{4} + 3461519512 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
71 $$( 1 + 823411004 T^{2} + 3069578968927394406 T^{4} + 823411004 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
73 $$( 1 - 4148991508 T^{2} + 8650887997979942790 T^{4} - 4148991508 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
79 $$( 1 + 2847963996 T^{2} + 15373907451301926406 T^{4} + 2847963996 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
83 $$( 1 + 7032005368 T^{2} + 39846447958779448850 T^{4} + 7032005368 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
89 $$( 1 + 57412 T + 10662063350 T^{2} + 57412 p^{5} T^{3} + p^{10} T^{4} )^{8}$$
97 $$( 1 - 31837059828 T^{2} +$$$$40\!\cdots\!94$$$$T^{4} - 31837059828 p^{10} T^{6} + p^{20} T^{8} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}