Properties

Label 16-768e8-1.1-c3e8-0-2
Degree $16$
Conductor $1.210\times 10^{23}$
Sign $1$
Analytic cond. $1.77753\times 10^{13}$
Root an. cond. $6.73152$
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  − 84·9-s + 152·25-s + 1.54e3·49-s + 3.37e3·73-s + 3.83e3·81-s − 8.56e3·97-s + 7.19e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.48e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 1.27e4·225-s + ⋯
L(s)  = 1  − 3.11·9-s + 1.21·25-s + 4.50·49-s + 5.41·73-s + 5.25·81-s − 8.96·97-s + 5.40·121-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 + 6.76·169-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 − 3.78·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{64} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.77753\times 10^{13}\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{64} \cdot 3^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(8.501670810\)
\(L(\frac12)\) \(\approx\) \(8.501670810\)
\(L(\frac{5}{2})\) 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 \( ( 1 + 14 p T^{2} + p^{6} T^{4} )^{2} \)
good5 \( ( 1 - 38 T^{2} + p^{6} T^{4} )^{4} \)
7 \( ( 1 - 386 T^{2} + p^{6} T^{4} )^{4} \)
11 \( ( 1 - 1798 T^{2} + p^{6} T^{4} )^{4} \)
13 \( ( 1 - 22 p^{2} T^{2} + p^{6} T^{4} )^{4} \)
17 \( ( 1 - 5218 T^{2} + p^{6} T^{4} )^{4} \)
19 \( ( 1 + 2186 T^{2} + p^{6} T^{4} )^{4} \)
23 \( ( 1 - 6770 T^{2} + p^{6} T^{4} )^{4} \)
29 \( ( 1 + 48490 T^{2} + p^{6} T^{4} )^{4} \)
31 \( ( 1 - 58610 T^{2} + p^{6} T^{4} )^{4} \)
37 \( ( 1 - 58870 T^{2} + p^{6} T^{4} )^{4} \)
41 \( ( 1 - 44530 T^{2} + p^{6} T^{4} )^{4} \)
43 \( ( 1 + 150266 T^{2} + p^{6} T^{4} )^{4} \)
47 \( ( 1 + 193822 T^{2} + p^{6} T^{4} )^{4} \)
53 \( ( 1 + 295162 T^{2} + p^{6} T^{4} )^{4} \)
59 \( ( 1 - 98854 T^{2} + p^{6} T^{4} )^{4} \)
61 \( ( 1 - 376678 T^{2} + p^{6} T^{4} )^{4} \)
67 \( ( 1 - 191062 T^{2} + p^{6} T^{4} )^{4} \)
71 \( ( 1 + 712366 T^{2} + p^{6} T^{4} )^{4} \)
73 \( ( 1 - 422 T + p^{3} T^{2} )^{8} \)
79 \( ( 1 - 539090 T^{2} + p^{6} T^{4} )^{4} \)
83 \( ( 1 - 1142710 T^{2} + p^{6} T^{4} )^{4} \)
89 \( ( 1 - 1270546 T^{2} + p^{6} T^{4} )^{4} \)
97 \( ( 1 + 1070 T + p^{3} T^{2} )^{8} \)
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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

−3.94442633369084435905572428154, −3.87953960702055116451319943245, −3.86079985609701950302077965268, −3.84776768930313278600699137768, −3.25920201254246930586983957242, −3.13515683055774566902171627809, −3.04298939373504140131705535964, −3.02712982249250521792361057842, −2.95662048030366176778689802953, −2.87677449541586876233952757864, −2.58179947070366455719163306291, −2.53678145110148456379847488193, −2.45729519506395321709430386301, −2.05560068235830371915387722960, −1.85614044847558988350764716628, −1.81970400463637963625862868593, −1.77716979605161011314923304723, −1.69457924893985888846738198097, −1.11221382676105538071999358852, −0.856156428748142263018517803663, −0.77121828348279385657970228768, −0.73055635460855957614974207674, −0.49367028645149247083518213242, −0.47710884118354095856484804550, −0.19035141056870628824150390883, 0.19035141056870628824150390883, 0.47710884118354095856484804550, 0.49367028645149247083518213242, 0.73055635460855957614974207674, 0.77121828348279385657970228768, 0.856156428748142263018517803663, 1.11221382676105538071999358852, 1.69457924893985888846738198097, 1.77716979605161011314923304723, 1.81970400463637963625862868593, 1.85614044847558988350764716628, 2.05560068235830371915387722960, 2.45729519506395321709430386301, 2.53678145110148456379847488193, 2.58179947070366455719163306291, 2.87677449541586876233952757864, 2.95662048030366176778689802953, 3.02712982249250521792361057842, 3.04298939373504140131705535964, 3.13515683055774566902171627809, 3.25920201254246930586983957242, 3.84776768930313278600699137768, 3.86079985609701950302077965268, 3.87953960702055116451319943245, 3.94442633369084435905572428154

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

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