Properties

Degree 16
Conductor $ 2^{24} \cdot 3^{8} \cdot 5^{16} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·16-s + 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 2·16-s + 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{24} \cdot 3^{8} \cdot 5^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{600} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{24} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3420173076\)
\(L(\frac12)\)  \(\approx\)  \(0.3420173076\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( 1 - T^{4} + T^{8} \)
5 \( 1 \)
good7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} )^{4} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T )^{8}( 1 + T )^{8} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{2} )^{8} \)
61 \( ( 1 - T )^{8}( 1 + T )^{8} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 - T^{4} + T^{8} )^{2} \)
79 \( ( 1 + T^{2} )^{8} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 - T^{2} + T^{4} )^{4} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−4.85792538980520456859987723705, −4.84279187873061813366787269754, −4.84021912317408969478257054777, −4.69713918049670469401304889610, −4.29461748777089127943875183278, −4.08650844877286557769597642430, −4.00331564109542064884171016440, −3.98257983847515790569620352270, −3.90200853071741372853193173707, −3.79227514851019030460906859615, −3.60742329880847449449973536253, −3.34758466400627477626343574810, −3.04905299221340423687937919644, −3.00787396981305638761693367608, −2.88852821851012515310168268816, −2.76206245112357439677472686171, −2.55498299986526963949050251362, −2.39104588902355201003887907027, −2.13344339935663428547069227099, −2.04197648193072394133171450053, −1.91540564081001050958220064208, −1.55063002561232208260280616514, −1.47224940725126572218738898426, −1.11543928520314058932352548067, −0.794170604188252699665396914671, 0.794170604188252699665396914671, 1.11543928520314058932352548067, 1.47224940725126572218738898426, 1.55063002561232208260280616514, 1.91540564081001050958220064208, 2.04197648193072394133171450053, 2.13344339935663428547069227099, 2.39104588902355201003887907027, 2.55498299986526963949050251362, 2.76206245112357439677472686171, 2.88852821851012515310168268816, 3.00787396981305638761693367608, 3.04905299221340423687937919644, 3.34758466400627477626343574810, 3.60742329880847449449973536253, 3.79227514851019030460906859615, 3.90200853071741372853193173707, 3.98257983847515790569620352270, 4.00331564109542064884171016440, 4.08650844877286557769597642430, 4.29461748777089127943875183278, 4.69713918049670469401304889610, 4.84021912317408969478257054777, 4.84279187873061813366787269754, 4.85792538980520456859987723705

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

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