Properties

 Degree $16$ Conductor $1.431\times 10^{21}$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 + 2·4-s + 67·16-s − 232·25-s + 64·37-s + 2.16e3·43-s + 324·64-s + 5.39e3·79-s − 464·100-s − 5.95e3·121-s + 127-s + 131-s + 137-s + 139-s + 128·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7.04e3·169-s + 4.32e3·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
 L(s)  = 1 + 1/4·4-s + 1.04·16-s − 1.85·25-s + 0.284·37-s + 7.66·43-s + 0.632·64-s + 7.67·79-s − 0.463·100-s − 4.47·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.0710·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 3.20·169-s + 1.91·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$16$$ Conductor: $$3^{16} \cdot 7^{16}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{441} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 3^{16} \cdot 7^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )$$

Particular Values

 $$L(2)$$ $$\approx$$ $$19.43053551$$ $$L(\frac12)$$ $$\approx$$ $$19.43053551$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$( 1 - T^{2} - p^{5} T^{4} - p^{6} T^{6} + p^{12} T^{8} )^{2}$$
5 $$( 1 + 116 T^{2} + 1282 p^{2} T^{4} + 116 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
11 $$( 1 + 2976 T^{2} + 47390 p^{2} T^{4} + 2976 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
13 $$( 1 + 3520 T^{2} + 5818162 T^{4} + 3520 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
17 $$( 1 + 9156 T^{2} + 49826306 T^{4} + 9156 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
19 $$( 1 + 4356 T^{2} + 98743142 T^{4} + 4356 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
23 $$( 1 + 42656 T^{2} + 750952798 T^{4} + 42656 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
29 $$( 1 + 44468 T^{2} + 996014662 T^{4} + 44468 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
31 $$( 1 + 99700 T^{2} + 4216698262 T^{4} + 99700 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
37 $$( 1 - 16 T + 9066 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
41 $$( 1 + 42420 T^{2} + 6386007458 T^{4} + 42420 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
43 $$( 1 - 540 T + 167814 T^{2} - 540 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
47 $$( 1 + 80164 T^{2} + 14237262198 T^{4} + 80164 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
53 $$( 1 + 462132 T^{2} + 97706926838 T^{4} + 462132 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
59 $$( 1 + 590420 T^{2} + 168915151078 T^{4} + 590420 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
61 $$( 1 + 162080 T^{2} - 25534056078 T^{4} + 162080 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
67 $$( 1 + 345126 T^{2} + p^{6} T^{4} )^{4}$$
71 $$( 1 + 472256 T^{2} + 269755485790 T^{4} + 472256 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
73 $$( 1 + 1020400 T^{2} + 546326225122 T^{4} + 1020400 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
79 $$( 1 - 1348 T + 16642 p T^{2} - 1348 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
83 $$( 1 + 367916 T^{2} + 118793240278 T^{4} + 367916 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
89 $$( 1 + 2588100 T^{2} + 2668496393378 T^{4} + 2588100 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
97 $$( 1 + 2268304 T^{2} + 2510084445826 T^{4} + 2268304 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$