Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 36·4-s + 120·8-s + 330·16-s + 8·23-s + 792·32-s + 64·46-s + 12·49-s + 8·53-s + 1.71e3·64-s + 48·79-s + 288·92-s + 96·98-s + 64·106-s − 16·107-s − 48·113-s + 80·121-s + 127-s + 3.43e3·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 384·158-s + 163-s + ⋯
L(s)  = 1  + 5.65·2-s + 18·4-s + 42.4·8-s + 82.5·16-s + 1.66·23-s + 140.·32-s + 9.43·46-s + 12/7·49-s + 1.09·53-s + 214.5·64-s + 5.40·79-s + 30.0·92-s + 9.69·98-s + 6.21·106-s − 1.54·107-s − 4.51·113-s + 7.27·121-s + 0.0887·127-s + 303.·128-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 + 30.5·158-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $837.5004687$
$L(\frac12)$  $\approx$  $837.5004687$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 - T )^{8} \)
3 \( 1 \)
5 \( 1 \)
7 \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
good11 \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 + 22 T^{2} + 259 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 16 T^{2} - 14 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - T + p T^{2} )^{8} \)
29 \( ( 1 - 30 T^{2} + 107 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 34 T^{2} + 1411 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 74 T^{2} + 3931 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 118 T^{2} + 6979 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 128 T^{2} + 7714 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 46 T^{2} + 5691 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 214 T^{2} + 18691 T^{4} - 214 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - p T^{2} )^{8} \)
71 \( ( 1 - 20 T^{2} - 2618 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 72 T^{2} + 4754 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 12 T + 144 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 62 T^{2} + 9739 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 116 T^{2} + 6406 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 48 T^{2} - 9406 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.70871632664595425332734892366, −3.63716920881122647887215034727, −3.37411254737377531120504563631, −3.30265025529125945214350900673, −3.19299183418908177482572634952, −3.14587158515161499169781840281, −2.97936886245568907935143633043, −2.90746544979422295273135663843, −2.80454516223016457762985062587, −2.73967783969234922599297110963, −2.38274476121712974568918724487, −2.24992848327993913588066033395, −2.20759258051920248665733107007, −2.16826248395092346880336251176, −2.12094465379866934711623394134, −2.10180467581432643270710542935, −2.00435793117497569964636886977, −1.40505562246421168005337828059, −1.34991033559655165765495886466, −1.26112194292300668730058430506, −1.19539301993949270472342004900, −1.07736682844406260445390677440, −0.811350011978743316061453134382, −0.56370170012592663241497367554, −0.24824282207563417968294495053, 0.24824282207563417968294495053, 0.56370170012592663241497367554, 0.811350011978743316061453134382, 1.07736682844406260445390677440, 1.19539301993949270472342004900, 1.26112194292300668730058430506, 1.34991033559655165765495886466, 1.40505562246421168005337828059, 2.00435793117497569964636886977, 2.10180467581432643270710542935, 2.12094465379866934711623394134, 2.16826248395092346880336251176, 2.20759258051920248665733107007, 2.24992848327993913588066033395, 2.38274476121712974568918724487, 2.73967783969234922599297110963, 2.80454516223016457762985062587, 2.90746544979422295273135663843, 2.97936886245568907935143633043, 3.14587158515161499169781840281, 3.19299183418908177482572634952, 3.30265025529125945214350900673, 3.37411254737377531120504563631, 3.63716920881122647887215034727, 3.70871632664595425332734892366

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

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