Properties

 Degree 32 Conductor $2^{96} \cdot 3^{32} \cdot 7^{16}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

Origins of factors

Dirichlet series

 L(s)  = 1 − 40·25-s − 8·49-s − 80·73-s − 112·97-s + 136·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-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 + ⋯
 L(s)  = 1 − 8·25-s − 8/7·49-s − 9.36·73-s − 11.3·97-s + 12.3·121-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 + 9.84·169-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 + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$32$$ $$N$$ = $$2^{96} \cdot 3^{32} \cdot 7^{16}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{4032} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(32,\ 2^{96} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )$ $L(1)$ $\approx$ $7.578069941e-5$ $L(\frac12)$ $\approx$ $7.578069941e-5$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7\}$, $$F_p$$ is a polynomial of degree 32. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 31.
$p$$F_p$
bad2 $$1$$
3 $$1$$
7 $$( 1 + T^{2} )^{8}$$
good5 $$( 1 + 2 p T^{2} + 54 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{4}$$
11 $$( 1 - 34 T^{2} + 510 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
13 $$( 1 - 32 T^{2} + 510 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
17 $$( 1 - 22 T^{2} + 174 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
19 $$( 1 + 10 T^{2} + p^{2} T^{4} )^{8}$$
23 $$( 1 + 70 T^{2} + 2262 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
29 $$( 1 - 8 T^{2} + 354 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
31 $$( 1 + 22 T^{2} + p^{2} T^{4} )^{8}$$
37 $$( 1 - 80 T^{2} + 3582 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
41 $$( 1 + 26 T^{2} + 3342 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
43 $$( 1 + 20 T^{2} - 1578 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
47 $$( 1 + 62 T^{2} + p^{2} T^{4} )^{8}$$
53 $$( 1 + 136 T^{2} + 8898 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
59 $$( 1 - p T^{2} )^{16}$$
61 $$( 1 - 14 T + p T^{2} )^{8}( 1 + 14 T + p T^{2} )^{8}$$
67 $$( 1 + 200 T^{2} + 18222 T^{4} + 200 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
71 $$( 1 + 254 T^{2} + 26022 T^{4} + 254 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
73 $$( 1 + 10 T + 150 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{8}$$
79 $$( 1 + 80 T^{2} + 7278 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
83 $$( 1 - 4 T^{2} + 5382 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
89 $$( 1 - 166 T^{2} + 22542 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
97 $$( 1 + 14 T + 222 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{8}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}