Properties

Label 16-12e24-1.1-c3e8-0-8
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $1.16755\times 10^{16}$
Root an. cond. $10.0972$
Motivic weight $3$
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  − 40·25-s + 2.06e3·49-s + 3.12e3·73-s − 3.41e3·97-s + 9.68e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.55e4·169-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 + ⋯
L(s)  = 1  − 0.319·25-s + 6.02·49-s + 5.01·73-s − 3.57·97-s + 7.27·121-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 + 7.07·169-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 + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(1.16755\times 10^{16}\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(60.74302225\)
\(L(\frac12)\) \(\approx\) \(60.74302225\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( ( 1 + 2 p T^{2} + p^{6} T^{4} )^{4} \)
7 \( ( 1 - 517 T^{2} + p^{6} T^{4} )^{4} \)
11 \( ( 1 - 2422 T^{2} + p^{6} T^{4} )^{4} \)
13 \( ( 1 - 7 p T + p^{3} T^{2} )^{4}( 1 + 7 p T + p^{3} T^{2} )^{4} \)
17 \( ( 1 - 9106 T^{2} + p^{6} T^{4} )^{4} \)
19 \( ( 1 + 7091 T^{2} + p^{6} T^{4} )^{4} \)
23 \( ( 1 + 23614 T^{2} + p^{6} T^{4} )^{4} \)
29 \( ( 1 + 14218 T^{2} + p^{6} T^{4} )^{4} \)
31 \( ( 1 - 29998 T^{2} + p^{6} T^{4} )^{4} \)
37 \( ( 1 - 100223 T^{2} + p^{6} T^{4} )^{4} \)
41 \( ( 1 - 65842 T^{2} + p^{6} T^{4} )^{4} \)
43 \( ( 1 + 120026 T^{2} + p^{6} T^{4} )^{4} \)
47 \( ( 1 + 120526 T^{2} + p^{6} T^{4} )^{4} \)
53 \( ( 1 + 20314 T^{2} + p^{6} T^{4} )^{4} \)
59 \( ( 1 + 75242 T^{2} + p^{6} T^{4} )^{4} \)
61 \( ( 1 - 233639 T^{2} + p^{6} T^{4} )^{4} \)
67 \( ( 1 + 601163 T^{2} + p^{6} T^{4} )^{4} \)
71 \( ( 1 + 367342 T^{2} + p^{6} T^{4} )^{4} \)
73 \( ( 1 - 391 T + p^{3} T^{2} )^{8} \)
79 \( ( 1 - 815509 T^{2} + p^{6} T^{4} )^{4} \)
83 \( ( 1 - 1128214 T^{2} + p^{6} T^{4} )^{4} \)
89 \( ( 1 + 318782 T^{2} + p^{6} T^{4} )^{4} \)
97 \( ( 1 + 427 T + p^{3} T^{2} )^{8} \)
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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

−3.60384016923051637599465042147, −3.33651767375785512328784329034, −3.16327302930514726005893983566, −3.10915235115189861687543476966, −3.03968027926798216046601839810, −2.88697848130741694990970970152, −2.76294498516896267703839154685, −2.72333871833256273564462987436, −2.56779062215656108152555684190, −2.20690998873861301001045797870, −2.20573528833785962101900327331, −2.12870468817607907956090860398, −1.98957016208659169448268239971, −1.82199994223844591826110100803, −1.75930977912992682704015611376, −1.58879736915747624994580697505, −1.43848341715490876283260790035, −1.20370022153543332330316575264, −0.857344083005708746432610614493, −0.819491260106450610304736498798, −0.72618684038995744833981885771, −0.62861314590456621352768003311, −0.45489948575800844242700943665, −0.40994099350387857002221462302, −0.33533880034637176823788594538, 0.33533880034637176823788594538, 0.40994099350387857002221462302, 0.45489948575800844242700943665, 0.62861314590456621352768003311, 0.72618684038995744833981885771, 0.819491260106450610304736498798, 0.857344083005708746432610614493, 1.20370022153543332330316575264, 1.43848341715490876283260790035, 1.58879736915747624994580697505, 1.75930977912992682704015611376, 1.82199994223844591826110100803, 1.98957016208659169448268239971, 2.12870468817607907956090860398, 2.20573528833785962101900327331, 2.20690998873861301001045797870, 2.56779062215656108152555684190, 2.72333871833256273564462987436, 2.76294498516896267703839154685, 2.88697848130741694990970970152, 3.03968027926798216046601839810, 3.10915235115189861687543476966, 3.16327302930514726005893983566, 3.33651767375785512328784329034, 3.60384016923051637599465042147

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

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