Properties

Degree 8
Conductor $ 3^{8} \cdot 7^{4} \cdot 11^{8} $
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  + 2·2-s + 2·5-s + 4·7-s − 4·8-s + 4·10-s − 10·13-s + 8·14-s − 6·16-s + 6·17-s − 18·19-s + 2·23-s − 12·25-s − 20·26-s + 6·29-s + 12·34-s + 8·35-s − 4·37-s − 36·38-s − 8·40-s − 20·43-s + 4·46-s + 6·47-s + 10·49-s − 24·50-s − 16·56-s + 12·58-s + 6·59-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.894·5-s + 1.51·7-s − 1.41·8-s + 1.26·10-s − 2.77·13-s + 2.13·14-s − 3/2·16-s + 1.45·17-s − 4.12·19-s + 0.417·23-s − 2.39·25-s − 3.92·26-s + 1.11·29-s + 2.05·34-s + 1.35·35-s − 0.657·37-s − 5.83·38-s − 1.26·40-s − 3.04·43-s + 0.589·46-s + 0.875·47-s + 10/7·49-s − 3.39·50-s − 2.13·56-s + 1.57·58-s + 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{8} \cdot 7^{4} \cdot 11^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  4
Selberg data  =  $(8,\ 3^{8} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + p^{2} T^{2} - p^{2} T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 16 T^{2} - 28 T^{3} + 111 T^{4} - 28 p T^{5} + 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 84 T^{2} + 422 T^{3} + 1844 T^{4} + 422 p T^{5} + 84 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 28 T^{2} - 24 T^{3} + 111 T^{4} - 24 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 184 T^{2} + 1278 T^{3} + 6468 T^{4} + 1278 p T^{5} + 184 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 28 T^{2} - 190 T^{3} + 516 T^{4} - 190 p T^{5} + 28 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 20 T^{2} - 126 T^{3} + 1404 T^{4} - 126 p T^{5} + 20 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 28 T^{2} - 216 T^{3} + 606 T^{4} - 216 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 72 T^{2} + 284 T^{3} + 4082 T^{4} + 284 p T^{5} + 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 124 T^{2} + 48 T^{3} + 6810 T^{4} + 48 p T^{5} + 124 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 282 T^{2} + 2632 T^{3} + 19943 T^{4} + 2632 p T^{5} + 282 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 44 T^{2} - 504 T^{3} + 93 p T^{4} - 504 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 148 T^{2} + 192 T^{3} + 9942 T^{4} + 192 p T^{5} + 148 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 208 T^{2} - 936 T^{3} + 17751 T^{4} - 936 p T^{5} + 208 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 108 T^{2} - 314 T^{3} + 3596 T^{4} - 314 p T^{5} + 108 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 130 T^{2} + 56 T^{3} + 8299 T^{4} + 56 p T^{5} + 130 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 220 T^{2} - 1014 T^{3} + 20916 T^{4} - 1014 p T^{5} + 220 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 34 T + 708 T^{2} + 9614 T^{3} + 96764 T^{4} + 9614 p T^{5} + 708 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 280 T^{2} + 1584 T^{3} + 8754 T^{4} + 1584 p T^{5} + 280 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 260 T^{2} - 1188 T^{3} + 30039 T^{4} - 1188 p T^{5} + 260 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 352 T^{2} + 4068 T^{3} + 48255 T^{4} + 4068 p T^{5} + 352 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 4 p T^{2} + 2758 T^{3} + 56428 T^{4} + 2758 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.85430533882383361643687803157, −5.61137777783146325121362489394, −5.35758966007766713949006977439, −5.31837359419914087547247198620, −5.14118472422754750640558160063, −4.88245444364321798527231414301, −4.69615157169450084801227010396, −4.66008900148433388667818920308, −4.60834196621281012460405749998, −4.29917916464032888887278041384, −3.96460653898983336389511226679, −3.94054620519324077956362912130, −3.89286044659267053587532369965, −3.51972012127801356651264369315, −3.36398273463782124607562027933, −3.00399513216261021161541558313, −2.54439667864073125802320061953, −2.48965236403515701653493404627, −2.43282157038605634055200820796, −2.41221243051754464504885238777, −2.00955101649199936892504172383, −1.81510793660135053301615531770, −1.36256341895812275294291747770, −1.29131866498753778410155147059, −1.17288776228353016214387945567, 0, 0, 0, 0, 1.17288776228353016214387945567, 1.29131866498753778410155147059, 1.36256341895812275294291747770, 1.81510793660135053301615531770, 2.00955101649199936892504172383, 2.41221243051754464504885238777, 2.43282157038605634055200820796, 2.48965236403515701653493404627, 2.54439667864073125802320061953, 3.00399513216261021161541558313, 3.36398273463782124607562027933, 3.51972012127801356651264369315, 3.89286044659267053587532369965, 3.94054620519324077956362912130, 3.96460653898983336389511226679, 4.29917916464032888887278041384, 4.60834196621281012460405749998, 4.66008900148433388667818920308, 4.69615157169450084801227010396, 4.88245444364321798527231414301, 5.14118472422754750640558160063, 5.31837359419914087547247198620, 5.35758966007766713949006977439, 5.61137777783146325121362489394, 5.85430533882383361643687803157

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