Properties

Degree 8
Conductor $ 2^{8} \cdot 3^{4} \cdot 67^{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  + 4·3-s − 8·5-s + 6·7-s + 6·9-s − 2·11-s + 18·13-s − 32·15-s − 6·17-s − 6·19-s + 24·21-s − 18·23-s + 20·25-s − 4·27-s + 6·29-s − 6·31-s − 8·33-s − 48·35-s − 10·37-s + 72·39-s − 2·41-s − 48·45-s − 6·47-s + 15·49-s − 24·51-s + 24·53-s + 16·55-s − 24·57-s + ⋯
L(s)  = 1  + 2.30·3-s − 3.57·5-s + 2.26·7-s + 2·9-s − 0.603·11-s + 4.99·13-s − 8.26·15-s − 1.45·17-s − 1.37·19-s + 5.23·21-s − 3.75·23-s + 4·25-s − 0.769·27-s + 1.11·29-s − 1.07·31-s − 1.39·33-s − 8.11·35-s − 1.64·37-s + 11.5·39-s − 0.312·41-s − 7.15·45-s − 0.875·47-s + 15/7·49-s − 3.36·51-s + 3.29·53-s + 2.15·55-s − 3.17·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 3^{4} \cdot 67^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{804} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{8} \cdot 3^{4} \cdot 67^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $3.29977$
$L(\frac12)$  $\approx$  $3.29977$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;67\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
7$D_4\times C_2$ \( 1 - 6 T + 3 p T^{2} - 54 T^{3} + 116 T^{4} - 54 p T^{5} + 3 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 10 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 2 T + p T^{2} )^{2} \)
17$D_4\times C_2$ \( 1 + 6 T + 31 T^{2} + 114 T^{3} + 276 T^{4} + 114 p T^{5} + 31 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 18 T + 163 T^{2} + 990 T^{3} + 4980 T^{4} + 990 p T^{5} + 163 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 6 T + 23 T^{2} - 66 T^{3} - 372 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$D_4\times C_2$ \( 1 + 10 T + 25 T^{2} + 10 T^{3} + 556 T^{4} + 10 p T^{5} + 25 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + 2 T - 73 T^{2} - 10 T^{3} + 4084 T^{4} - 10 p T^{5} - 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 12 T^{2} - 2410 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 6 T + 107 T^{2} + 570 T^{3} + 7380 T^{4} + 570 p T^{5} + 107 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
59$D_4\times C_2$ \( 1 - 196 T^{2} + 16182 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 18 T + 225 T^{2} - 2106 T^{3} + 16556 T^{4} - 2106 p T^{5} + 225 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 30 T + 499 T^{2} - 5970 T^{3} + 55860 T^{4} - 5970 p T^{5} + 499 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 2 T - 119 T^{2} + 46 T^{3} + 9508 T^{4} + 46 p T^{5} - 119 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 18 T + 261 T^{2} + 2754 T^{3} + 25700 T^{4} + 2754 p T^{5} + 261 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 6 T + p T^{2} + 426 T^{3} - 852 T^{4} + 426 p T^{5} + p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 76 T^{2} + 3462 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 18 T + 257 T^{2} - 2682 T^{3} + 23268 T^{4} - 2682 p T^{5} + 257 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.52704863177309229765839636218, −7.34791836989052170388315673152, −7.25410243528187087338243911908, −6.69619877314121659852713035107, −6.58383774324913178166195040662, −6.32364909425578328219476656322, −5.81394908606086645678508820606, −5.74917165277786606635523765439, −5.72313604307986687771539784995, −5.01761970851814346606948405147, −4.95319984068026430352370400162, −4.21355485292172450076571675996, −4.19633567698243252583469471942, −4.04683176555877781936001640893, −3.98759498332930193505150608106, −3.72743960242981800705260923880, −3.51374106422397149560284641196, −3.48618449586255284097732342985, −3.10452491063081726081917585052, −2.08872438393129253203191730676, −2.08165071876625340917275967084, −2.05437118195364086643866712957, −1.85586136177714784575576231631, −0.874407742149730763175277766597, −0.46308863549140384750307188062, 0.46308863549140384750307188062, 0.874407742149730763175277766597, 1.85586136177714784575576231631, 2.05437118195364086643866712957, 2.08165071876625340917275967084, 2.08872438393129253203191730676, 3.10452491063081726081917585052, 3.48618449586255284097732342985, 3.51374106422397149560284641196, 3.72743960242981800705260923880, 3.98759498332930193505150608106, 4.04683176555877781936001640893, 4.19633567698243252583469471942, 4.21355485292172450076571675996, 4.95319984068026430352370400162, 5.01761970851814346606948405147, 5.72313604307986687771539784995, 5.74917165277786606635523765439, 5.81394908606086645678508820606, 6.32364909425578328219476656322, 6.58383774324913178166195040662, 6.69619877314121659852713035107, 7.25410243528187087338243911908, 7.34791836989052170388315673152, 7.52704863177309229765839636218

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