Properties

Label 16-12e24-1.1-c1e8-0-2
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $1.31391\times 10^{9}$
Root an. cond. $3.71458$
Motivic weight $1$
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  − 6·5-s + 2·13-s + 5·25-s + 6·29-s + 8·37-s − 24·41-s − 19·49-s + 2·61-s − 12·65-s + 4·73-s + 4·97-s + 66·101-s − 16·109-s − 30·113-s + 32·121-s + 42·125-s + 127-s + 131-s + 137-s + 139-s − 36·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 37·169-s + ⋯
L(s)  = 1  − 2.68·5-s + 0.554·13-s + 25-s + 1.11·29-s + 1.31·37-s − 3.74·41-s − 2.71·49-s + 0.256·61-s − 1.48·65-s + 0.468·73-s + 0.406·97-s + 6.56·101-s − 1.53·109-s − 2.82·113-s + 2.90·121-s + 3.75·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.84·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(1.31391\times 10^{9}\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5661829925\)
\(L(\frac12)\) \(\approx\) \(0.5661829925\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( ( 1 + 3 T + 11 T^{2} + 24 T^{3} + 54 T^{4} + 24 p T^{5} + 11 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
7 \( 1 + 19 T^{2} + 181 T^{4} + 1558 T^{6} + 12310 T^{8} + 1558 p^{2} T^{10} + 181 p^{4} T^{12} + 19 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 32 T^{2} + 559 T^{4} - 7136 T^{6} + 77680 T^{8} - 7136 p^{2} T^{10} + 559 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - T - 17 T^{2} + 8 T^{3} + 142 T^{4} + 8 p T^{5} - 17 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 61 T^{2} + 1500 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 49 T^{2} + 1248 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 - 77 T^{2} + 3397 T^{4} - 113498 T^{6} + 2994742 T^{8} - 113498 p^{2} T^{10} + 3397 p^{4} T^{12} - 77 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 3 T + 59 T^{2} - 168 T^{3} + 2382 T^{4} - 168 p T^{5} + 59 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 55 T^{2} + 1345 T^{4} - 13310 T^{6} - 1008146 T^{8} - 13310 p^{2} T^{10} + 1345 p^{4} T^{12} + 55 p^{6} T^{14} + p^{8} T^{16} \)
37 \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 12 T + 131 T^{2} + 996 T^{3} + 7176 T^{4} + 996 p T^{5} + 131 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 + 64 T^{2} - 593 T^{4} + 63424 T^{6} + 10994416 T^{8} + 63424 p^{2} T^{10} - 593 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 53 T^{2} + 2053 T^{4} + 194086 T^{6} - 10513226 T^{8} + 194086 p^{2} T^{10} + 2053 p^{4} T^{12} - 53 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 136 T^{2} + 9054 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( 1 - 56 T^{2} - 617 T^{4} + 179704 T^{6} - 8948768 T^{8} + 179704 p^{2} T^{10} - 617 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 8 p T^{5} - 113 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 160 T^{2} + 10255 T^{4} + 1018720 T^{6} + 100399504 T^{8} + 1018720 p^{2} T^{10} + 10255 p^{4} T^{12} + 160 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 140 T^{2} + 10230 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - T + 138 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \)
79 \( 1 + 115 T^{2} - 2555 T^{4} + 379270 T^{6} + 127521094 T^{8} + 379270 p^{2} T^{10} - 2555 p^{4} T^{12} + 115 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 221 T^{2} + 22861 T^{4} - 2696642 T^{6} + 291036430 T^{8} - 2696642 p^{2} T^{10} + 22861 p^{4} T^{12} - 221 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 184 T^{2} + 21006 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 2 T - 59 T^{2} + 262 T^{3} - 5828 T^{4} + 262 p T^{5} - 59 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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

−3.88074580891553822842346961778, −3.74098737810386314294192071421, −3.60771864916944807005549439080, −3.59059415980102795349685980664, −3.49342858486640232749479682849, −3.32742528744028794654925122170, −3.23905772667244709055868097554, −3.17131780330014694596399757639, −3.02408282208190283879432884505, −2.90677093891733064306820097722, −2.66786393125838756279043477595, −2.54911171869896736911343563463, −2.34720720293436557848111278739, −2.26608696023373612616145897765, −2.02905897925084089394173579564, −1.86230638476041415293179096631, −1.68099078736303798302124313777, −1.55765566820182753069189586846, −1.51178308547729397445461668758, −1.48589992828108610432835399879, −0.77737210546552802990792038105, −0.69708161300708096054089383316, −0.69339152863933710561095272778, −0.46203918409998538061410207575, −0.10876514460939150405045194650, 0.10876514460939150405045194650, 0.46203918409998538061410207575, 0.69339152863933710561095272778, 0.69708161300708096054089383316, 0.77737210546552802990792038105, 1.48589992828108610432835399879, 1.51178308547729397445461668758, 1.55765566820182753069189586846, 1.68099078736303798302124313777, 1.86230638476041415293179096631, 2.02905897925084089394173579564, 2.26608696023373612616145897765, 2.34720720293436557848111278739, 2.54911171869896736911343563463, 2.66786393125838756279043477595, 2.90677093891733064306820097722, 3.02408282208190283879432884505, 3.17131780330014694596399757639, 3.23905772667244709055868097554, 3.32742528744028794654925122170, 3.49342858486640232749479682849, 3.59059415980102795349685980664, 3.60771864916944807005549439080, 3.74098737810386314294192071421, 3.88074580891553822842346961778

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

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