Properties

Label 16-12e24-1.1-c1e8-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·13-s + 8·25-s + 24·37-s + 28·49-s − 8·61-s + 24·73-s + 56·97-s − 48·109-s − 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2.21·13-s + 8/5·25-s + 3.94·37-s + 4·49-s − 1.02·61-s + 2.80·73-s + 5.68·97-s − 4.59·109-s − 5.09·121-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 − 1.53·169-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 + ⋯

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

\(L(1)\) \(\approx\) \(1.742455371\)
\(L(\frac12)\) \(\approx\) \(1.742455371\)
\(L(\frac{3}{2})\) 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 \( ( 1 - 4 T^{2} + 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
7 \( ( 1 - 2 p T^{2} + 99 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 28 T^{2} + 390 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 20 T^{2} + 726 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 52 T^{2} + 1590 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 68 T^{2} + 2310 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 100 T^{2} + 5094 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 76 T^{2} + 4662 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 20 T^{2} - 1194 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 28 T^{2} + 4806 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 2 p T^{2} + 12699 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 62 T^{2} + 2643 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 268 T^{2} + 30966 T^{4} + 268 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 308 T^{2} + 39126 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 7 T + p T^{2} )^{8} \)
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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

−4.17089813260078709832147692753, −3.73220579314558851810244513061, −3.62363841132530049623370407540, −3.53886239796710554584173185250, −3.43781919945944616354705328845, −3.42245376212811949010983934546, −3.04366525766328825292717262962, −2.97748860867132492979997480728, −2.85571357245303933713055038882, −2.74227088568524671898105096105, −2.61232254629017445791751252265, −2.59449880349643795372450009826, −2.32168510592406414465441140464, −2.24814175625428053578526816389, −2.12999037466114420816594087130, −2.11561151717147986201816741154, −1.79994655036323348597449368478, −1.73006087471869684524593458834, −1.33379954911521026932377881978, −1.04967275144514586173511710710, −0.961122673316914609062951652607, −0.952719425213923681364360251676, −0.860863024718210153044497222022, −0.43267851944749187253922365517, −0.12956246680418703808014698963, 0.12956246680418703808014698963, 0.43267851944749187253922365517, 0.860863024718210153044497222022, 0.952719425213923681364360251676, 0.961122673316914609062951652607, 1.04967275144514586173511710710, 1.33379954911521026932377881978, 1.73006087471869684524593458834, 1.79994655036323348597449368478, 2.11561151717147986201816741154, 2.12999037466114420816594087130, 2.24814175625428053578526816389, 2.32168510592406414465441140464, 2.59449880349643795372450009826, 2.61232254629017445791751252265, 2.74227088568524671898105096105, 2.85571357245303933713055038882, 2.97748860867132492979997480728, 3.04366525766328825292717262962, 3.42245376212811949010983934546, 3.43781919945944616354705328845, 3.53886239796710554584173185250, 3.62363841132530049623370407540, 3.73220579314558851810244513061, 4.17089813260078709832147692753

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

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