Properties

Degree 8
Conductor $ 3^{8} \cdot 11^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 7·4-s + 5-s − 3·7-s + 15·8-s + 3·10-s − 9·11-s − 9·13-s − 9·14-s + 30·16-s − 2·17-s − 10·19-s + 7·20-s − 27·22-s + 4·23-s − 27·26-s − 21·28-s + 10·29-s + 8·31-s + 57·32-s − 6·34-s − 3·35-s − 3·37-s − 30·38-s + 15·40-s − 23·41-s + 16·43-s + ⋯
L(s)  = 1  + 2.12·2-s + 7/2·4-s + 0.447·5-s − 1.13·7-s + 5.30·8-s + 0.948·10-s − 2.71·11-s − 2.49·13-s − 2.40·14-s + 15/2·16-s − 0.485·17-s − 2.29·19-s + 1.56·20-s − 5.75·22-s + 0.834·23-s − 5.29·26-s − 3.96·28-s + 1.85·29-s + 1.43·31-s + 10.0·32-s − 1.02·34-s − 0.507·35-s − 0.493·37-s − 4.86·38-s + 2.37·40-s − 3.59·41-s + 2.43·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(96059601\)    =    \(3^{8} \cdot 11^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{99} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 96059601,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.94519$
$L(\frac12)$  $\approx$  $2.94519$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
11$C_4$ \( 1 + 9 T + 41 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
good2$C_2^2:C_4$ \( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2^2:C_4$ \( 1 - T + T^{2} - 11 T^{3} + 36 T^{4} - 11 p T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$C_4\times C_2$ \( 1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 15 p T^{5} + 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 + 9 T + 18 T^{2} - 115 T^{3} - 789 T^{4} - 115 p T^{5} + 18 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 + 2 T - 13 T^{2} + 20 T^{3} + 341 T^{4} + 20 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 70 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_4\times C_2$ \( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 200 p T^{5} + 31 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^2:C_4$ \( 1 - 8 T + 3 T^{2} - 46 T^{3} + 1175 T^{4} - 46 p T^{5} + 3 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 155 p T^{5} - 18 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 + 23 T + 208 T^{2} + 961 T^{3} + 3975 T^{4} + 961 p T^{5} + 208 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 - 3 T - 43 T^{2} + 45 T^{3} + 2116 T^{4} + 45 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 + 6 T + 23 T^{2} - 120 T^{3} - 1319 T^{4} - 120 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 - 20 T + 131 T^{2} - 530 T^{3} + 3851 T^{4} - 530 p T^{5} + 131 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2:C_4$ \( 1 - 3 T - 7 T^{2} - 441 T^{3} + 4900 T^{4} - 441 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2:C_4$ \( 1 - 27 T + 253 T^{2} - 819 T^{3} + 100 T^{4} - 819 p T^{5} + 253 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 - 6 T - 57 T^{2} + 130 T^{3} + 4761 T^{4} + 130 p T^{5} - 57 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 - 5 T + 6 T^{2} - 715 T^{3} + 9821 T^{4} - 715 p T^{5} + 6 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 + 21 T + 88 T^{2} - 915 T^{3} - 13199 T^{4} - 915 p T^{5} + 88 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 10 T + 183 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2:C_4$ \( 1 + 33 T + 537 T^{2} + 6655 T^{3} + 71196 T^{4} + 6655 p T^{5} + 537 p^{2} T^{6} + 33 p^{3} T^{7} + p^{4} T^{8} \)
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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

−10.51361359276888462709231394160, −10.21656361328303341069277793850, −9.796024811089212381660377861908, −9.706824278428610577373155902898, −9.535510910893011390947133214239, −8.465345796978454265012026473221, −8.250019655275872751043527251637, −8.215873627560383150253341548128, −8.077741216665055473900571049257, −7.08585544143943675140507218754, −7.03175841904256586093581601287, −6.95837837767768320332728360660, −6.87911670244084392931576428502, −6.13731191355275439606919306151, −5.86792800328731985688302816538, −5.57709364750440716429837695488, −5.00212064571773703624658333110, −4.96092440515681747857783805520, −4.69391520472387922687432997586, −4.24605824971914652230569860549, −3.70461679290982177190831233442, −2.99109732830900938047165688224, −2.67353381586304762334914147412, −2.44875023367782687263432536837, −2.17771360057825058654361214910, 2.17771360057825058654361214910, 2.44875023367782687263432536837, 2.67353381586304762334914147412, 2.99109732830900938047165688224, 3.70461679290982177190831233442, 4.24605824971914652230569860549, 4.69391520472387922687432997586, 4.96092440515681747857783805520, 5.00212064571773703624658333110, 5.57709364750440716429837695488, 5.86792800328731985688302816538, 6.13731191355275439606919306151, 6.87911670244084392931576428502, 6.95837837767768320332728360660, 7.03175841904256586093581601287, 7.08585544143943675140507218754, 8.077741216665055473900571049257, 8.215873627560383150253341548128, 8.250019655275872751043527251637, 8.465345796978454265012026473221, 9.535510910893011390947133214239, 9.706824278428610577373155902898, 9.796024811089212381660377861908, 10.21656361328303341069277793850, 10.51361359276888462709231394160

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