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  − 4·4-s + 2·9-s + 8·13-s + 12·16-s + 16·25-s − 8·36-s + 32·37-s + 20·49-s − 32·52-s − 24·61-s − 32·64-s + 40·73-s − 5·81-s − 24·97-s − 64·100-s + 24·109-s + 16·117-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 24·144-s − 128·148-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 2·4-s + 2/3·9-s + 2.21·13-s + 3·16-s + 16/5·25-s − 4/3·36-s + 5.26·37-s + 20/7·49-s − 4.43·52-s − 3.07·61-s − 4·64-s + 4.68·73-s − 5/9·81-s − 2.43·97-s − 6.39·100-s + 2.29·109-s + 1.47·117-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s − 10.5·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-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  =  $(8,\ 2^{8} \cdot 3^{4} \cdot 67^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $3.28845$
$L(\frac12)$  $\approx$  $3.28845$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 100 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
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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

−7.63963407630896693890836180260, −7.06940000786716704339351723361, −6.99028103061584082549561968000, −6.49911245637268527186132071557, −6.48074209884614893404768362839, −6.12042785738433281686824664655, −5.95736697000494878546597121092, −5.80437583900437488492381753774, −5.43175626954294552499861907331, −5.31584971960212456468378711703, −4.73589338871446679481855148646, −4.65753866281560205616606545555, −4.59070768999280769743554230802, −4.31094421622386009377705044832, −4.03903389093982803431406964090, −3.63680614981927736974151880905, −3.60731602762211980599705831166, −3.27396045296346793535723177972, −2.79696675267356136779345135149, −2.60670422944462861464638233188, −2.31273361621025530140938294406, −1.45990207639139647581672111987, −1.15243595717591120963763979206, −0.892092873694361216457947448404, −0.74290849436228213523786050826, 0.74290849436228213523786050826, 0.892092873694361216457947448404, 1.15243595717591120963763979206, 1.45990207639139647581672111987, 2.31273361621025530140938294406, 2.60670422944462861464638233188, 2.79696675267356136779345135149, 3.27396045296346793535723177972, 3.60731602762211980599705831166, 3.63680614981927736974151880905, 4.03903389093982803431406964090, 4.31094421622386009377705044832, 4.59070768999280769743554230802, 4.65753866281560205616606545555, 4.73589338871446679481855148646, 5.31584971960212456468378711703, 5.43175626954294552499861907331, 5.80437583900437488492381753774, 5.95736697000494878546597121092, 6.12042785738433281686824664655, 6.48074209884614893404768362839, 6.49911245637268527186132071557, 6.99028103061584082549561968000, 7.06940000786716704339351723361, 7.63963407630896693890836180260

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