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

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 20·25-s − 24·43-s − 2·49-s + 48·67-s − 8·73-s − 40·97-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 4·25-s − 3.65·43-s − 2/7·49-s + 5.86·67-s − 0.936·73-s − 4.06·97-s + 8/11·121-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 − 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-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  =  $(8,\ 2^{24} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $0.5695052350$
$L(\frac12)$  $\approx$  $0.5695052350$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.87472715297778390839506675012, −5.79252410295276734172617003258, −5.69141735556805144807283262511, −5.25287280722824407309816137669, −5.20548079281903520116489852706, −5.06279258406415346869821531979, −4.86784729279691208672959611053, −4.58598802393403868999780784986, −4.24225296431010947518190985975, −4.08020597178771150561004771456, −4.00705067315927215019143456888, −3.62116990776835524145782518484, −3.61746058591613972159331306048, −3.40449678285312499520422442216, −3.16931585724086181994128067915, −2.77943286531179220805733196495, −2.59418935690332892199898002178, −2.21319651930375439917683443574, −2.10110872134390969165841895213, −1.86315077615667525680878371579, −1.72993651638594484156621580269, −1.23445644086025279572211032659, −1.11474261491534579866124691249, −0.46918899259827702462309735149, −0.14228804874971602617736121744, 0.14228804874971602617736121744, 0.46918899259827702462309735149, 1.11474261491534579866124691249, 1.23445644086025279572211032659, 1.72993651638594484156621580269, 1.86315077615667525680878371579, 2.10110872134390969165841895213, 2.21319651930375439917683443574, 2.59418935690332892199898002178, 2.77943286531179220805733196495, 3.16931585724086181994128067915, 3.40449678285312499520422442216, 3.61746058591613972159331306048, 3.62116990776835524145782518484, 4.00705067315927215019143456888, 4.08020597178771150561004771456, 4.24225296431010947518190985975, 4.58598802393403868999780784986, 4.86784729279691208672959611053, 5.06279258406415346869821531979, 5.20548079281903520116489852706, 5.25287280722824407309816137669, 5.69141735556805144807283262511, 5.79252410295276734172617003258, 5.87472715297778390839506675012

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