Properties

Degree 8
Conductor $ 2^{8} \cdot 3^{4} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 6·9-s − 40·13-s − 48·16-s + 340·25-s + 24·36-s − 520·37-s + 1.25e3·49-s + 160·52-s − 1.76e3·61-s + 448·64-s + 1.64e3·73-s − 693·81-s + 3.08e3·97-s − 1.36e3·100-s − 4.26e3·109-s + 240·117-s − 2.92e3·121-s + 127-s + 131-s + 137-s + 139-s + 288·144-s + 2.08e3·148-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1/2·4-s − 2/9·9-s − 0.853·13-s − 3/4·16-s + 2.71·25-s + 1/9·36-s − 2.31·37-s + 3.65·49-s + 0.426·52-s − 3.71·61-s + 7/8·64-s + 2.62·73-s − 0.950·81-s + 3.22·97-s − 1.35·100-s − 3.74·109-s + 0.189·117-s − 2.19·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 1/6·144-s + 1.15·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(20736\)    =    \(2^{8} \cdot 3^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{12} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 20736,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)$
$L(2)$  $\approx$  $0.650064$
$L(\frac12)$  $\approx$  $0.650064$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2$C_2^2$ \( 1 + p^{2} T^{2} + p^{6} T^{4} \)
3$C_2^2$ \( 1 + 2 p T^{2} + p^{6} T^{4} \)
good5$C_2^2$ \( ( 1 - 34 p T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 626 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 1462 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 10 T + p^{3} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 8546 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 8858 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 14926 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 25658 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 9122 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 130 T + p^{3} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 122162 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 108554 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 170014 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 74 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 380758 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 442 T + p^{3} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 60026 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 364178 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 410 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 978818 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 428954 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 703058 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 770 T + p^{3} T^{2} )^{4} \)
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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

−15.21922772648573504637430320642, −14.75833816652542838209323584236, −14.14962300175953305361265734246, −14.14424290558261059727321999050, −13.61236202681288569750388567473, −13.33606714947912092173490198699, −12.74846519852760496323923348560, −12.30827082394096717786522346978, −12.13622797696963170452285217551, −11.78638494336544608530849049629, −10.90848303310514202961708056084, −10.63583717773865906397103834563, −10.53836900377380684539559835725, −9.765570252606697516390225853530, −9.171279628502259032815510915885, −8.826431911939698895359179639394, −8.710033085418608479691832702699, −7.79450003352070673570454316107, −7.25318987538709189929739072122, −6.84183802607566256049597967972, −6.19761641252313747646880668290, −5.11970921045757825472644203448, −4.99667685049821096552388374399, −3.96721015998628628573876114770, −2.75108982179113338574342640129, 2.75108982179113338574342640129, 3.96721015998628628573876114770, 4.99667685049821096552388374399, 5.11970921045757825472644203448, 6.19761641252313747646880668290, 6.84183802607566256049597967972, 7.25318987538709189929739072122, 7.79450003352070673570454316107, 8.710033085418608479691832702699, 8.826431911939698895359179639394, 9.171279628502259032815510915885, 9.765570252606697516390225853530, 10.53836900377380684539559835725, 10.63583717773865906397103834563, 10.90848303310514202961708056084, 11.78638494336544608530849049629, 12.13622797696963170452285217551, 12.30827082394096717786522346978, 12.74846519852760496323923348560, 13.33606714947912092173490198699, 13.61236202681288569750388567473, 14.14424290558261059727321999050, 14.14962300175953305361265734246, 14.75833816652542838209323584236, 15.21922772648573504637430320642

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