Properties

Degree 8
Conductor $ 2^{4} \cdot 3^{8} \cdot 5^{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  − 16·13-s − 16-s + 8·17-s + 8·23-s + 8·29-s + 24·31-s − 8·37-s + 4·43-s + 20·47-s − 16·53-s − 28·67-s + 8·83-s − 24·89-s + 24·97-s + 24·103-s − 32·107-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + ⋯
L(s)  = 1  − 4.43·13-s − 1/4·16-s + 1.94·17-s + 1.66·23-s + 1.48·29-s + 4.31·31-s − 1.31·37-s + 0.609·43-s + 2.91·47-s − 2.19·53-s − 3.42·67-s + 0.878·83-s − 2.54·89-s + 2.43·97-s + 2.36·103-s − 3.09·107-s + 2.90·121-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 + 9.84·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $1.543084021$
$L(\frac12)$  $\approx$  $1.543084021$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ \( 1 + T^{4} \)
3 \( 1 \)
5 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good11$D_4\times C_2$ \( 1 - 32 T^{2} + 466 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 168 T^{3} + 866 T^{4} - 168 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 120 T^{3} + 386 T^{4} - 120 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} + 72 T^{3} - 622 T^{4} + 72 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 76 T^{2} + 3654 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} + 220 T^{3} - 3554 T^{4} + 220 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 20 T + 200 T^{2} - 1220 T^{3} + 7246 T^{4} - 1220 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 16 T + 128 T^{2} + 1296 T^{3} + 12338 T^{4} + 1296 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 28 T + 392 T^{2} + 4508 T^{3} + 43006 T^{4} + 4508 p T^{5} + 392 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 236 T^{2} + 23494 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^3$ \( 1 - 8542 T^{4} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 600 T^{3} + 11186 T^{4} - 600 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 12 T + 206 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 24 T + 288 T^{2} - 3960 T^{3} + 49826 T^{4} - 3960 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.11388273049151311751016322272, −6.07452319785739702541658928777, −5.96556388729189918560825686561, −5.27652924173928646324344956071, −5.14464916689989158402647429335, −5.09235137449360328006964745444, −5.01938473869581091185097159526, −4.68064805368551518299335524531, −4.66484310720473843057625193760, −4.47258105463508237614342821531, −4.23038619623491200332514704539, −3.91179107788775218385828811412, −3.58106677386434096493193452698, −3.17303382254405491354622127774, −3.17122841109996786359978097928, −2.74330350723390561401731630015, −2.67356263157165126108825506957, −2.66455459517936266052932394310, −2.33687392539710334173977200568, −2.12895179207047191317585245611, −1.46005722063724427455159793663, −1.36138840676851781010874336715, −0.963382300773776048038622673884, −0.71873921477873437996708461310, −0.21224958927974796384427538446, 0.21224958927974796384427538446, 0.71873921477873437996708461310, 0.963382300773776048038622673884, 1.36138840676851781010874336715, 1.46005722063724427455159793663, 2.12895179207047191317585245611, 2.33687392539710334173977200568, 2.66455459517936266052932394310, 2.67356263157165126108825506957, 2.74330350723390561401731630015, 3.17122841109996786359978097928, 3.17303382254405491354622127774, 3.58106677386434096493193452698, 3.91179107788775218385828811412, 4.23038619623491200332514704539, 4.47258105463508237614342821531, 4.66484310720473843057625193760, 4.68064805368551518299335524531, 5.01938473869581091185097159526, 5.09235137449360328006964745444, 5.14464916689989158402647429335, 5.27652924173928646324344956071, 5.96556388729189918560825686561, 6.07452319785739702541658928777, 6.11388273049151311751016322272

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