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  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.9381679553$
$L(\frac12)$  $\approx$  $0.9381679553$
$L(\frac{3}{2})$   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

−3.61261201988250633661110299621, −3.19743576660060655518440465008, −3.13311808277714411674218311228, −3.12440634316891798124181334725, −3.03925044760253783128288857005, −2.83216607829951677981930043383, −2.72796970284675827340773650657, −2.69294376489547866044025259639, −2.53881126697437707771702740197, −2.18018595585389559316627035009, −2.12739128706677589010360492127, −2.10663081065998814485596459061, −1.99097960309358159841943461110, −1.93633265305773165420542068486, −1.84485130205414414187123624013, −1.59567589643450197558645628472, −1.53998293753685243949603595295, −1.37111067517537875049375871336, −1.21819689321840552581469706288, −0.76125672152594412186332497541, −0.68926245233958698564802213843, −0.66876321119493066531613599678, −0.64857798360169942970717899194, −0.37439406263420404213465104189, −0.35889591257301730401713236085, 0.35889591257301730401713236085, 0.37439406263420404213465104189, 0.64857798360169942970717899194, 0.66876321119493066531613599678, 0.68926245233958698564802213843, 0.76125672152594412186332497541, 1.21819689321840552581469706288, 1.37111067517537875049375871336, 1.53998293753685243949603595295, 1.59567589643450197558645628472, 1.84485130205414414187123624013, 1.93633265305773165420542068486, 1.99097960309358159841943461110, 2.10663081065998814485596459061, 2.12739128706677589010360492127, 2.18018595585389559316627035009, 2.53881126697437707771702740197, 2.69294376489547866044025259639, 2.72796970284675827340773650657, 2.83216607829951677981930043383, 3.03925044760253783128288857005, 3.12440634316891798124181334725, 3.13311808277714411674218311228, 3.19743576660060655518440465008, 3.61261201988250633661110299621

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

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