Properties

Label 16-48e16-1.1-c1e8-0-1
Degree $16$
Conductor $7.941\times 10^{26}$
Sign $1$
Analytic cond. $1.31243\times 10^{10}$
Root an. cond. $4.28923$
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 + 40·37-s − 56·61-s − 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 2.21·13-s + 6.57·37-s − 7.17·61-s − 0.766·109-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 + 2.46·169-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{64} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(1.31243\times 10^{10}\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{64} \cdot 3^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.182008024\)
\(L(\frac12)\) \(\approx\) \(1.182008024\)
\(L(\frac{3}{2})\) 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 \( ( 1 + p^{2} T^{4} )^{4} \)
7 \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + p^{2} T^{4} )^{4} \)
13 \( ( 1 - 2 T + p T^{2} )^{4}( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17 \( ( 1 + p T^{2} )^{8} \)
19 \( ( 1 - 46 T^{4} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - p T^{2} )^{8} \)
29 \( ( 1 + p^{2} T^{4} )^{4} \)
31 \( ( 1 + 194 T^{4} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
41 \( ( 1 - p T^{2} )^{8} \)
43 \( ( 1 - 3214 T^{4} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + p T^{2} )^{8} \)
53 \( ( 1 + p^{2} T^{4} )^{4} \)
59 \( ( 1 + p^{2} T^{4} )^{4} \)
61 \( ( 1 + 14 T + p T^{2} )^{4}( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \)
67 \( ( 1 + 5906 T^{4} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - p T^{2} )^{8} \)
73 \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \)
79 \( ( 1 + 7682 T^{4} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + p^{2} T^{4} )^{4} \)
89 \( ( 1 - p T^{2} )^{8} \)
97 \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \)
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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

−3.76989677159229344038473931681, −3.66243766781013881886716977819, −3.48698054325897862424510666638, −3.39612916553051431304691875738, −3.18219709544077357349458994975, −3.10562037065779567244079280093, −3.08438486079658086701868200140, −2.99020854895387734454766143423, −2.77142668409070737687800926766, −2.62214359484035452841674696322, −2.58787161712220285390004411847, −2.44610968035787209939317576470, −2.27753846910265299087436406223, −2.23835924680964445355596804284, −1.74239218752982577622006665020, −1.67061549212658918453612669832, −1.63357317184786226376672506814, −1.62177224685869915673908173932, −1.49240085889563123572732993797, −1.12497876184753900303032513162, −0.888487536748921548333230445150, −0.822868703384004403576106687549, −0.808282490628075205722926757131, −0.51284004097641588083151319081, −0.07244231811309053405953560284, 0.07244231811309053405953560284, 0.51284004097641588083151319081, 0.808282490628075205722926757131, 0.822868703384004403576106687549, 0.888487536748921548333230445150, 1.12497876184753900303032513162, 1.49240085889563123572732993797, 1.62177224685869915673908173932, 1.63357317184786226376672506814, 1.67061549212658918453612669832, 1.74239218752982577622006665020, 2.23835924680964445355596804284, 2.27753846910265299087436406223, 2.44610968035787209939317576470, 2.58787161712220285390004411847, 2.62214359484035452841674696322, 2.77142668409070737687800926766, 2.99020854895387734454766143423, 3.08438486079658086701868200140, 3.10562037065779567244079280093, 3.18219709544077357349458994975, 3.39612916553051431304691875738, 3.48698054325897862424510666638, 3.66243766781013881886716977819, 3.76989677159229344038473931681

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

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