Properties

Degree 16
Conductor $ 2^{24} \cdot 3^{8} \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  − 2·4-s + 8·7-s + 8·9-s − 5·16-s + 32·25-s − 16·28-s − 16·36-s + 12·49-s + 64·63-s + 20·64-s − 80·79-s + 30·81-s − 64·100-s − 40·112-s − 40·121-s + 127-s + 131-s + 137-s + 139-s − 40·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 64·169-s + 173-s + ⋯
L(s)  = 1  − 4-s + 3.02·7-s + 8/3·9-s − 5/4·16-s + 32/5·25-s − 3.02·28-s − 8/3·36-s + 12/7·49-s + 8.06·63-s + 5/2·64-s − 9.00·79-s + 10/3·81-s − 6.39·100-s − 3.77·112-s − 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.33·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.92·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{24} \cdot 3^{8} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{168} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{24} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(2.89364\)
\(L(\frac12)\)  \(\approx\)  \(2.89364\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \)
3 \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
7 \( ( 1 - 2 T + p T^{2} )^{4} \)
good5 \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 + p T^{2} )^{8} \)
19 \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \)
37 \( ( 1 - p T^{2} )^{8} \)
41 \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 + p T^{2} )^{8} \)
59 \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + 10 T + p T^{2} )^{8} \)
83 \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 + p T^{2} )^{8} \)
97 \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{4} \)
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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

−5.70900835221872772173934723285, −5.66451042194735201129528785452, −5.49725246135198787427274720579, −5.39142267679829561118612244354, −4.95793576867031106065680223806, −4.94522912674983041577673598076, −4.79457472372657306113581235709, −4.71718816802015742626785186484, −4.68690920803570552858768595132, −4.41750777975697444234847449990, −4.32076100637348721087153216491, −4.27271080404863649582117291601, −4.22733816769738465197811367851, −3.66417686544298282379807499248, −3.57090048957578998834165564553, −3.18807791239472889795321765492, −3.00992148967276773330653814579, −2.80169869128384198928605592664, −2.76039720214996990031075185995, −2.21259845014422628275842591264, −2.05455184383410879499229456875, −1.63677850635857086974171701745, −1.38987445863716406825841116413, −1.23722799363660427789192439325, −1.08963056607366955327383653957, 1.08963056607366955327383653957, 1.23722799363660427789192439325, 1.38987445863716406825841116413, 1.63677850635857086974171701745, 2.05455184383410879499229456875, 2.21259845014422628275842591264, 2.76039720214996990031075185995, 2.80169869128384198928605592664, 3.00992148967276773330653814579, 3.18807791239472889795321765492, 3.57090048957578998834165564553, 3.66417686544298282379807499248, 4.22733816769738465197811367851, 4.27271080404863649582117291601, 4.32076100637348721087153216491, 4.41750777975697444234847449990, 4.68690920803570552858768595132, 4.71718816802015742626785186484, 4.79457472372657306113581235709, 4.94522912674983041577673598076, 4.95793576867031106065680223806, 5.39142267679829561118612244354, 5.49725246135198787427274720579, 5.66451042194735201129528785452, 5.70900835221872772173934723285

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

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