Properties

Label 16-384e8-1.1-c3e8-0-1
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $6.94349\times 10^{10}$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 84·9-s + 56·25-s − 424·49-s − 784·73-s + 3.83e3·81-s + 7.95e3·97-s + 8.29e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.12e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 4.70e3·225-s + ⋯
L(s)  = 1  + 28/9·9-s + 0.447·25-s − 1.23·49-s − 1.25·73-s + 5.25·81-s + 8.32·97-s + 6.23·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4.15·169-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 + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 1.39·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(6.94349\times 10^{10}\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(2.405697029\)
\(L(\frac12)\) \(\approx\) \(2.405697029\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - 14 p T^{2} + p^{6} T^{4} )^{2} \)
good5 \( ( 1 - 14 T^{2} + p^{6} T^{4} )^{4} \)
7 \( ( 1 + 106 T^{2} + p^{6} T^{4} )^{4} \)
11 \( ( 1 - 2074 T^{2} + p^{6} T^{4} )^{4} \)
13 \( ( 1 - 2282 T^{2} + p^{6} T^{4} )^{4} \)
17 \( ( 1 - 3554 T^{2} + p^{6} T^{4} )^{4} \)
19 \( ( 1 + 13622 T^{2} + p^{6} T^{4} )^{4} \)
23 \( ( 1 - 1010 T^{2} + p^{6} T^{4} )^{4} \)
29 \( ( 1 + 35842 T^{2} + p^{6} T^{4} )^{4} \)
31 \( ( 1 - 20774 T^{2} + p^{6} T^{4} )^{4} \)
37 \( ( 1 + 2182 T^{2} + p^{6} T^{4} )^{4} \)
41 \( ( 1 - 37490 T^{2} + p^{6} T^{4} )^{4} \)
43 \( ( 1 - 71482 T^{2} + p^{6} T^{4} )^{4} \)
47 \( ( 1 + p^{3} T^{2} )^{8} \)
53 \( ( 1 + 181330 T^{2} + p^{6} T^{4} )^{4} \)
59 \( ( 1 + 54950 T^{2} + p^{6} T^{4} )^{4} \)
61 \( ( 1 - 451850 T^{2} + p^{6} T^{4} )^{4} \)
67 \( ( 1 + 483926 T^{2} + p^{6} T^{4} )^{4} \)
71 \( ( 1 - 526034 T^{2} + p^{6} T^{4} )^{4} \)
73 \( ( 1 + 98 T + p^{3} T^{2} )^{8} \)
79 \( ( 1 - 966278 T^{2} + p^{6} T^{4} )^{4} \)
83 \( ( 1 - 288106 T^{2} + p^{6} T^{4} )^{4} \)
89 \( ( 1 - 1253138 T^{2} + p^{6} T^{4} )^{4} \)
97 \( ( 1 - 994 T + p^{3} T^{2} )^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.66616206273111467967187853302, −4.52890727365823970342674741125, −4.28082258940131946800791950233, −4.18365302683294084410152948034, −3.98953100121802425398018391921, −3.65357994608837598693893439459, −3.53540455629913678050352828471, −3.49248484065519108073886674256, −3.45123977817075166112358756222, −3.28349870213977647556580859059, −3.16902937763608299507832379788, −2.70783414673740442347743630272, −2.64380087907602283372871893536, −2.39979652365768792667051287686, −2.16502019171112792806219164201, −2.01863805771051213509993075156, −1.94111512362755127075359867654, −1.70126419417642958783379305254, −1.56441666068763428868856933088, −1.30249833285139632846898695969, −1.05912019949094372486020196405, −0.877228435168249603757270141702, −0.64859244481819706043041688458, −0.61047732943306169778364558116, −0.084592198364477537566414641403, 0.084592198364477537566414641403, 0.61047732943306169778364558116, 0.64859244481819706043041688458, 0.877228435168249603757270141702, 1.05912019949094372486020196405, 1.30249833285139632846898695969, 1.56441666068763428868856933088, 1.70126419417642958783379305254, 1.94111512362755127075359867654, 2.01863805771051213509993075156, 2.16502019171112792806219164201, 2.39979652365768792667051287686, 2.64380087907602283372871893536, 2.70783414673740442347743630272, 3.16902937763608299507832379788, 3.28349870213977647556580859059, 3.45123977817075166112358756222, 3.49248484065519108073886674256, 3.53540455629913678050352828471, 3.65357994608837598693893439459, 3.98953100121802425398018391921, 4.18365302683294084410152948034, 4.28082258940131946800791950233, 4.52890727365823970342674741125, 4.66616206273111467967187853302

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.