Properties

Degree 8
Conductor $ 2^{12} \cdot 3^{4} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·4-s − 8·8-s + 6·9-s − 32·11-s − 4·16-s − 8·17-s + 12·18-s + 32·19-s − 64·22-s + 28·25-s − 8·32-s − 16·34-s − 12·36-s + 64·38-s + 40·41-s + 32·43-s + 64·44-s + 76·49-s + 56·50-s − 128·59-s − 24·64-s − 256·67-s + 16·68-s − 48·72-s + 200·73-s − 64·76-s + ⋯
L(s)  = 1  + 2-s − 1/2·4-s − 8-s + 2/3·9-s − 2.90·11-s − 1/4·16-s − 0.470·17-s + 2/3·18-s + 1.68·19-s − 2.90·22-s + 1.11·25-s − 1/4·32-s − 0.470·34-s − 1/3·36-s + 1.68·38-s + 0.975·41-s + 0.744·43-s + 1.45·44-s + 1.55·49-s + 1.11·50-s − 2.16·59-s − 3/8·64-s − 3.82·67-s + 4/17·68-s − 2/3·72-s + 2.73·73-s − 0.842·76-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(331776\)    =    \(2^{12} \cdot 3^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{24} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 331776,\ (\ :1, 1, 1, 1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.883563$
$L(\frac12)$  $\approx$  $0.883563$
$L(2)$   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$D_{4}$ \( 1 - p T + 3 p T^{2} - p^{3} T^{3} + p^{4} T^{4} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2^2 \wr C_2$ \( 1 - 28 T^{2} + 678 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} \)
7$C_2^2 \wr C_2$ \( 1 - 76 T^{2} + 3174 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{4} \)
13$C_2^2 \wr C_2$ \( 1 - 292 T^{2} + 75366 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} \)
17$D_{4}$ \( ( 1 + 4 T + 390 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 16 T + 738 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 1636 T^{2} + 1179654 T^{4} - 1636 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - 1756 T^{2} + 1539558 T^{4} - 1756 p^{4} T^{6} + p^{8} T^{8} \)
31$C_2^2 \wr C_2$ \( 1 - 460 T^{2} - 683610 T^{4} - 460 p^{4} T^{6} + p^{8} T^{8} \)
37$C_2^2 \wr C_2$ \( 1 - 4612 T^{2} + 8955366 T^{4} - 4612 p^{4} T^{6} + p^{8} T^{8} \)
41$D_{4}$ \( ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 16 T + 3330 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - 5284 T^{2} + 13593798 T^{4} - 5284 p^{4} T^{6} + p^{8} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - 9436 T^{2} + 38033574 T^{4} - 9436 p^{4} T^{6} + p^{8} T^{8} \)
59$D_{4}$ \( ( 1 + 64 T + 7554 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
61$C_2^2 \wr C_2$ \( 1 - 11332 T^{2} + 56649510 T^{4} - 11332 p^{4} T^{6} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 + 128 T + 12642 T^{2} + 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - 11236 T^{2} + 75307398 T^{4} - 11236 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 100 T + 10086 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^2 \wr C_2$ \( 1 - 17548 T^{2} + 151931046 T^{4} - 17548 p^{4} T^{6} + p^{8} T^{8} \)
83$D_{4}$ \( ( 1 - 80 T + 14610 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 100 T + 11430 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 28 T + 12102 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−13.38007144419951117460720073910, −13.01856626614660602989844745783, −12.63206147962328451585534254910, −12.40726944006768489398401158527, −12.09426334571972855301389625462, −11.74012896545958677224446927740, −10.80626742570984102236577363067, −10.76293102666116434721923976255, −10.70464897985039189470434307052, −10.19749434703203209978610563842, −9.496348007137509508991465712168, −9.284699578591150270908648951262, −9.110724250132546308985061125581, −8.293193361109677564047267773112, −7.82621438104503750076123410916, −7.77757615597159208351556357411, −7.16093454727449572360219821350, −6.74789168780566424944579060693, −5.78138868873153761470703667285, −5.55622303442588941650820156498, −5.14059384744607714226052879422, −4.61158122069529405690835623875, −4.24261628617026933892896956709, −3.19126847826640312997642897644, −2.68155907806404530091965479096, 2.68155907806404530091965479096, 3.19126847826640312997642897644, 4.24261628617026933892896956709, 4.61158122069529405690835623875, 5.14059384744607714226052879422, 5.55622303442588941650820156498, 5.78138868873153761470703667285, 6.74789168780566424944579060693, 7.16093454727449572360219821350, 7.77757615597159208351556357411, 7.82621438104503750076123410916, 8.293193361109677564047267773112, 9.110724250132546308985061125581, 9.284699578591150270908648951262, 9.496348007137509508991465712168, 10.19749434703203209978610563842, 10.70464897985039189470434307052, 10.76293102666116434721923976255, 10.80626742570984102236577363067, 11.74012896545958677224446927740, 12.09426334571972855301389625462, 12.40726944006768489398401158527, 12.63206147962328451585534254910, 13.01856626614660602989844745783, 13.38007144419951117460720073910

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