Properties

Degree 8
Conductor $ 2^{24} \cdot 3^{8} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 8·17-s + 12·25-s − 16·31-s + 8·41-s − 16·47-s + 10·49-s − 16·71-s + 24·73-s + 16·79-s + 8·89-s + 8·97-s + 16·103-s + 40·113-s + 32·119-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.94·17-s + 12/5·25-s − 2.87·31-s + 1.24·41-s − 2.33·47-s + 10/7·49-s − 1.89·71-s + 2.80·73-s + 1.80·79-s + 0.847·89-s + 0.812·97-s + 1.57·103-s + 3.76·113-s + 2.93·119-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.307·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{24} \cdot 3^{8} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4032} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{24} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $1.949473440$
$L(\frac12)$  $\approx$  $1.949473440$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
good5$D_4\times C_2$ \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 + 4 T^{2} + 42 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
19$D_4\times C_2$ \( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 84 T^{2} + 3254 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 116 T^{2} + 5910 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
43$D_4\times C_2$ \( 1 - 140 T^{2} + 8406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 84 T^{2} + 8426 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 52 T^{2} + 7530 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 164 T^{2} + 17802 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.22346244067086025248847766358, −5.76350657422559029789120680018, −5.62037114378331800433256324955, −5.30950037888657056992897584542, −5.12068952021409522697721662617, −4.98155777499815363662452736756, −4.75226825423893273235075553954, −4.72753801948510179414020189152, −4.20620152303346507327336569959, −4.19023035457148739337713994622, −4.06816711069393023174207906793, −3.52535459676349387984507876790, −3.39418598480175409499983283713, −3.39314592152252274967408894875, −3.18347499421894553522735376704, −2.94947552660839131081949625188, −2.54175496718756926735722025659, −2.29918942516118968933742609884, −2.16341679187750412422639067135, −1.87035801753806654882220904304, −1.79502251677854224753786018614, −1.06119790628535093346913823740, −1.03903121379069734055922913777, −0.40776064959521645545769169560, −0.34492076556418182734895864822, 0.34492076556418182734895864822, 0.40776064959521645545769169560, 1.03903121379069734055922913777, 1.06119790628535093346913823740, 1.79502251677854224753786018614, 1.87035801753806654882220904304, 2.16341679187750412422639067135, 2.29918942516118968933742609884, 2.54175496718756926735722025659, 2.94947552660839131081949625188, 3.18347499421894553522735376704, 3.39314592152252274967408894875, 3.39418598480175409499983283713, 3.52535459676349387984507876790, 4.06816711069393023174207906793, 4.19023035457148739337713994622, 4.20620152303346507327336569959, 4.72753801948510179414020189152, 4.75226825423893273235075553954, 4.98155777499815363662452736756, 5.12068952021409522697721662617, 5.30950037888657056992897584542, 5.62037114378331800433256324955, 5.76350657422559029789120680018, 6.22346244067086025248847766358

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