Properties

Degree 8
Conductor $ 2^{16} \cdot 3^{8} \cdot 5^{8} $
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  − 4·13-s + 4·37-s + 4·73-s + 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-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 + ⋯
L(s)  = 1  − 4·13-s + 4·37-s + 4·73-s + 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{16} \cdot 3^{8} \cdot 5^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{3600} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{16} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.9122044279\)
\(L(\frac12)\)  \(\approx\)  \(0.9122044279\)
\(L(1)\)   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 8. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.29596244471888632877575886245, −6.25443319809882618780658313562, −5.67988046107564714656975990578, −5.66979806296969947094178306581, −5.39124956076113367634690634621, −5.05493350352688407288564698729, −5.01615814203792437299528726007, −4.90449726426965376192210600265, −4.78952659669949514086963591717, −4.43822268827050100980762420462, −4.25824456212995958992485879422, −4.04770330587375452454865044459, −3.86226711840103339051260172007, −3.60535296727699148717555715829, −3.20289258474364028574680896810, −3.08710023255538321158036724516, −2.79469070822809598425280377879, −2.41360974141554376186838932514, −2.39260388633025422999209861023, −2.31431995466167475123812049611, −2.10734300108169582526941406054, −1.65719099457495338257020987190, −1.03496738874621367978271446123, −1.00610473011614275017122542910, −0.39850318103113549641196065899, 0.39850318103113549641196065899, 1.00610473011614275017122542910, 1.03496738874621367978271446123, 1.65719099457495338257020987190, 2.10734300108169582526941406054, 2.31431995466167475123812049611, 2.39260388633025422999209861023, 2.41360974141554376186838932514, 2.79469070822809598425280377879, 3.08710023255538321158036724516, 3.20289258474364028574680896810, 3.60535296727699148717555715829, 3.86226711840103339051260172007, 4.04770330587375452454865044459, 4.25824456212995958992485879422, 4.43822268827050100980762420462, 4.78952659669949514086963591717, 4.90449726426965376192210600265, 5.01615814203792437299528726007, 5.05493350352688407288564698729, 5.39124956076113367634690634621, 5.66979806296969947094178306581, 5.67988046107564714656975990578, 6.25443319809882618780658313562, 6.29596244471888632877575886245

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