Properties

Degree 8
Conductor $ 2^{4} \cdot 3^{8} \cdot 223^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 4

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 10·4-s − 2·5-s − 5·7-s + 20·8-s − 8·10-s − 4·11-s − 5·13-s − 20·14-s + 35·16-s + 2·17-s − 7·19-s − 20·20-s − 16·22-s + 4·23-s − 11·25-s − 20·26-s − 50·28-s − 9·29-s − 31-s + 56·32-s + 8·34-s + 10·35-s − 11·37-s − 28·38-s − 40·40-s − 14·41-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s − 0.894·5-s − 1.88·7-s + 7.07·8-s − 2.52·10-s − 1.20·11-s − 1.38·13-s − 5.34·14-s + 35/4·16-s + 0.485·17-s − 1.60·19-s − 4.47·20-s − 3.41·22-s + 0.834·23-s − 2.19·25-s − 3.92·26-s − 9.44·28-s − 1.67·29-s − 0.179·31-s + 9.89·32-s + 1.37·34-s + 1.69·35-s − 1.80·37-s − 4.54·38-s − 6.32·40-s − 2.18·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 223^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 223^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{4} \cdot 3^{8} \cdot 223^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4014} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  4
Selberg data  =  $(8,\ 2^{4} \cdot 3^{8} \cdot 223^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;223\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{2,\;3,\;223\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 - T )^{4} \)
3 \( 1 \)
223$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 2 T + 3 p T^{2} + 19 T^{3} + 97 T^{4} + 19 p T^{5} + 3 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 5 T + 30 T^{2} + 92 T^{3} + 310 T^{4} + 92 p T^{5} + 30 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 4 T + 31 T^{2} + 123 T^{3} + 456 T^{4} + 123 p T^{5} + 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 5 T + 34 T^{2} + 150 T^{3} + 654 T^{4} + 150 p T^{5} + 34 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 2 T + 25 T^{2} - 27 T^{3} + 270 T^{4} - 27 p T^{5} + 25 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 7 T + 48 T^{2} + 210 T^{3} + 998 T^{4} + 210 p T^{5} + 48 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 4 T + 35 T^{2} - 9 p T^{3} + 1066 T^{4} - 9 p^{2} T^{5} + 35 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 9 T + 4 p T^{2} + 626 T^{3} + 4734 T^{4} + 626 p T^{5} + 4 p^{3} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + T + 78 T^{2} + 118 T^{3} + 3036 T^{4} + 118 p T^{5} + 78 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 11 T + 155 T^{2} + 1129 T^{3} + 8591 T^{4} + 1129 p T^{5} + 155 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 14 T + 139 T^{2} + 1245 T^{3} + 9124 T^{4} + 1245 p T^{5} + 139 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 22 T + 343 T^{2} + 3395 T^{3} + 26448 T^{4} + 3395 p T^{5} + 343 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 8 T + 181 T^{2} - 1085 T^{3} + 12569 T^{4} - 1085 p T^{5} + 181 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 3 T + 146 T^{2} + 380 T^{3} + 10800 T^{4} + 380 p T^{5} + 146 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 6 T + 135 T^{2} - 553 T^{3} + 8832 T^{4} - 553 p T^{5} + 135 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 9 T + 236 T^{2} + 1484 T^{3} + 21094 T^{4} + 1484 p T^{5} + 236 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 22 T + 443 T^{2} + 5021 T^{3} + 51131 T^{4} + 5021 p T^{5} + 443 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 5 T + 215 T^{2} + 624 T^{3} + 19864 T^{4} + 624 p T^{5} + 215 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 3 T + 159 T^{2} + 527 T^{3} + 16559 T^{4} + 527 p T^{5} + 159 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 29 T + 523 T^{2} + 6871 T^{3} + 70149 T^{4} + 6871 p T^{5} + 523 p^{2} T^{6} + 29 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 12 T + 327 T^{2} - 2665 T^{3} + 40215 T^{4} - 2665 p T^{5} + 327 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 9 T + 186 T^{2} + 2446 T^{3} + 17334 T^{4} + 2446 p T^{5} + 186 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 29 T + 630 T^{2} + 8898 T^{3} + 102354 T^{4} + 8898 p T^{5} + 630 p^{2} T^{6} + 29 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.32783990413348826683985525312, −6.12619056656465431979251036947, −5.86821876256841211923549038090, −5.73945024892319749640850179828, −5.45004453513696317635018713083, −5.24689427694212538791892563048, −5.10882068020397855925354803669, −5.09953041377310457369382716618, −4.91432531589148299245896366837, −4.40459894800188114180195460974, −4.35672888197568198243809482881, −4.16206298161431499245167700811, −4.02950398793090092217253431249, −3.65585388575758733420682034485, −3.46452530421827535140746085749, −3.43642762080122250982860197875, −3.28182365885607572341236543046, −2.92678435209176391308625275471, −2.74070500089549069615216174380, −2.54419204484245494543720674953, −2.50046153014486675826209355010, −1.86095785676571554135156501400, −1.68481640021134302968970852400, −1.56051860456743620234937186886, −1.47426509677664598195120476195, 0, 0, 0, 0, 1.47426509677664598195120476195, 1.56051860456743620234937186886, 1.68481640021134302968970852400, 1.86095785676571554135156501400, 2.50046153014486675826209355010, 2.54419204484245494543720674953, 2.74070500089549069615216174380, 2.92678435209176391308625275471, 3.28182365885607572341236543046, 3.43642762080122250982860197875, 3.46452530421827535140746085749, 3.65585388575758733420682034485, 4.02950398793090092217253431249, 4.16206298161431499245167700811, 4.35672888197568198243809482881, 4.40459894800188114180195460974, 4.91432531589148299245896366837, 5.09953041377310457369382716618, 5.10882068020397855925354803669, 5.24689427694212538791892563048, 5.45004453513696317635018713083, 5.73945024892319749640850179828, 5.86821876256841211923549038090, 6.12619056656465431979251036947, 6.32783990413348826683985525312

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