Properties

Label 16-12e24-1.1-c2e8-0-3
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $2.41562\times 10^{13}$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·5-s − 8·13-s − 24·17-s − 56·25-s − 128·29-s − 24·37-s − 160·41-s + 124·49-s − 48·53-s + 136·61-s − 64·65-s + 72·73-s − 192·85-s + 168·89-s + 104·97-s − 352·101-s + 48·109-s − 728·113-s + 568·121-s − 744·125-s + 127-s + 131-s + 137-s + 139-s − 1.02e3·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 8/5·5-s − 0.615·13-s − 1.41·17-s − 2.23·25-s − 4.41·29-s − 0.648·37-s − 3.90·41-s + 2.53·49-s − 0.905·53-s + 2.22·61-s − 0.984·65-s + 0.986·73-s − 2.25·85-s + 1.88·89-s + 1.07·97-s − 3.48·101-s + 0.440·109-s − 6.44·113-s + 4.69·121-s − 5.95·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.06·145-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(2.41562\times 10^{13}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.06015887585\)
\(L(\frac12)\) \(\approx\) \(0.06015887585\)
\(L(2)\) 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 \( ( 1 - 4 T + 52 T^{2} - 124 T^{3} + 1558 T^{4} - 124 p^{2} T^{5} + 52 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
7 \( 1 - 124 T^{2} + 5002 T^{4} + 178832 T^{6} - 21308909 T^{8} + 178832 p^{4} T^{10} + 5002 p^{8} T^{12} - 124 p^{12} T^{14} + p^{16} T^{16} \)
11 \( 1 - 568 T^{2} + 14804 p T^{4} - 31597576 T^{6} + 4462882054 T^{8} - 31597576 p^{4} T^{10} + 14804 p^{9} T^{12} - 568 p^{12} T^{14} + p^{16} T^{16} \)
13 \( ( 1 + 4 T + 394 T^{2} - 272 T^{3} + 72883 T^{4} - 272 p^{2} T^{5} + 394 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( ( 1 + 12 T + 436 T^{2} + 7380 T^{3} + 90294 T^{4} + 7380 p^{2} T^{5} + 436 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
19 \( 1 - 1340 T^{2} + 1130122 T^{4} - 629721776 T^{6} + 266027004115 T^{8} - 629721776 p^{4} T^{10} + 1130122 p^{8} T^{12} - 1340 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 2872 T^{2} + 3949276 T^{4} - 3455093896 T^{6} + 2139369590086 T^{8} - 3455093896 p^{4} T^{10} + 3949276 p^{8} T^{12} - 2872 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 + 64 T + 2596 T^{2} + 67264 T^{3} + 1985254 T^{4} + 67264 p^{2} T^{5} + 2596 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 - 184 p T^{2} + 15692188 T^{4} - 26845223416 T^{6} + 31031256945478 T^{8} - 26845223416 p^{4} T^{10} + 15692188 p^{8} T^{12} - 184 p^{13} T^{14} + p^{16} T^{16} \)
37 \( ( 1 + 12 T + 2650 T^{2} - 36432 T^{3} + 2846211 T^{4} - 36432 p^{2} T^{5} + 2650 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( ( 1 + 80 T + 5956 T^{2} + 285680 T^{3} + 13442758 T^{4} + 285680 p^{2} T^{5} + 5956 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 - 12424 T^{2} + 71116252 T^{4} - 244515946552 T^{6} + 551660618273158 T^{8} - 244515946552 p^{4} T^{10} + 71116252 p^{8} T^{12} - 12424 p^{12} T^{14} + p^{16} T^{16} \)
47 \( 1 - 2872 T^{2} + 15358300 T^{4} - 35747158408 T^{6} + 102474993504454 T^{8} - 35747158408 p^{4} T^{10} + 15358300 p^{8} T^{12} - 2872 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 + 24 T + 7204 T^{2} + 159624 T^{3} + 28315302 T^{4} + 159624 p^{2} T^{5} + 7204 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 - 11320 T^{2} + 50642716 T^{4} - 127137029128 T^{6} + 320662122219526 T^{8} - 127137029128 p^{4} T^{10} + 50642716 p^{8} T^{12} - 11320 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 - 68 T + 8266 T^{2} - 137072 T^{3} + 22167667 T^{4} - 137072 p^{2} T^{5} + 8266 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 - 27292 T^{2} + 345823978 T^{4} - 2700513303280 T^{6} + 14433873700614067 T^{8} - 2700513303280 p^{4} T^{10} + 345823978 p^{8} T^{12} - 27292 p^{12} T^{14} + p^{16} T^{16} \)
71 \( 1 - 11080 T^{2} + 30779164 T^{4} + 199057866248 T^{6} - 1902994100012090 T^{8} + 199057866248 p^{4} T^{10} + 30779164 p^{8} T^{12} - 11080 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 - 36 T + 9130 T^{2} - 318096 T^{3} + 41836659 T^{4} - 318096 p^{2} T^{5} + 9130 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 41308 T^{2} + 786693994 T^{4} - 9045860371504 T^{6} + 68675576167144435 T^{8} - 9045860371504 p^{4} T^{10} + 786693994 p^{8} T^{12} - 41308 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 - 8200 T^{2} + 140721244 T^{4} - 1081724015032 T^{6} + 8821781005832710 T^{8} - 1081724015032 p^{4} T^{10} + 140721244 p^{8} T^{12} - 8200 p^{12} T^{14} + p^{16} T^{16} \)
89 \( ( 1 - 84 T + 22900 T^{2} - 1903500 T^{3} + 237150582 T^{4} - 1903500 p^{2} T^{5} + 22900 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 52 T + 22522 T^{2} - 1223152 T^{3} + 279148675 T^{4} - 1223152 p^{2} T^{5} + 22522 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.80763850091956630491590850470, −3.50827926572362293592985538428, −3.41803066173297462216340164345, −3.41084347102244334703632373978, −3.36722437340851111026292329038, −3.14275218271398358229483798098, −2.98253527947236713919839635130, −2.73950507979496945166901248542, −2.59187633604016889560388280094, −2.34782539293609483553473203083, −2.32464003110700278898184809727, −2.22843647115315784135964639149, −2.08386496587734030294022076838, −2.05693874097741981822762830370, −1.90841552446221222199871159667, −1.81870989066385292734661651505, −1.60117924682558629388917949967, −1.45796478984853238768485067938, −1.43042810732465880947062999073, −1.05835827692874315216956013028, −0.951974211211188949844246594516, −0.53837754499369090984439100671, −0.49100809763707120611175436827, −0.13973506312835965792146059153, −0.03879565427560685921012873050, 0.03879565427560685921012873050, 0.13973506312835965792146059153, 0.49100809763707120611175436827, 0.53837754499369090984439100671, 0.951974211211188949844246594516, 1.05835827692874315216956013028, 1.43042810732465880947062999073, 1.45796478984853238768485067938, 1.60117924682558629388917949967, 1.81870989066385292734661651505, 1.90841552446221222199871159667, 2.05693874097741981822762830370, 2.08386496587734030294022076838, 2.22843647115315784135964639149, 2.32464003110700278898184809727, 2.34782539293609483553473203083, 2.59187633604016889560388280094, 2.73950507979496945166901248542, 2.98253527947236713919839635130, 3.14275218271398358229483798098, 3.36722437340851111026292329038, 3.41084347102244334703632373978, 3.41803066173297462216340164345, 3.50827926572362293592985538428, 3.80763850091956630491590850470

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

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