Properties

Degree 8
Conductor $ 2^{8} \cdot 3^{12} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·5-s − 7-s + 36·11-s + 5·13-s + 2·19-s − 99·23-s + 22·25-s + 63·29-s − 7·31-s + 9·35-s − 64·37-s + 18·41-s − 46·43-s + 81·47-s + 24·49-s − 324·55-s − 126·59-s + 41·61-s − 45·65-s + 116·67-s + 86·73-s − 36·77-s + 83·79-s + 81·83-s − 5·91-s − 18·95-s − 196·97-s + ⋯
L(s)  = 1  − 9/5·5-s − 1/7·7-s + 3.27·11-s + 5/13·13-s + 2/19·19-s − 4.30·23-s + 0.879·25-s + 2.17·29-s − 0.225·31-s + 9/35·35-s − 1.72·37-s + 0.439·41-s − 1.06·43-s + 1.72·47-s + 0.489·49-s − 5.89·55-s − 2.13·59-s + 0.672·61-s − 0.692·65-s + 1.73·67-s + 1.17·73-s − 0.467·77-s + 1.05·79-s + 0.975·83-s − 0.0549·91-s − 0.189·95-s − 2.02·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 3^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{8} \cdot 3^{12} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.29185\)
\(L(\frac12)\)  \(\approx\)  \(1.29185\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 + 9 T + 59 T^{2} + 288 T^{3} + 1074 T^{4} + 288 p^{2} T^{5} + 59 p^{4} T^{6} + 9 p^{6} T^{7} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 + T - 23 T^{2} - 74 T^{3} - 1874 T^{4} - 74 p^{2} T^{5} - 23 p^{4} T^{6} + p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 - 36 T + 683 T^{2} - 9036 T^{3} + 100632 T^{4} - 9036 p^{2} T^{5} + 683 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} \)
13$D_4\times C_2$ \( 1 - 5 T - 245 T^{2} + 340 T^{3} + 40114 T^{4} + 340 p^{2} T^{5} - 245 p^{4} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 769 T^{2} + 298176 T^{4} - 769 p^{4} T^{6} + p^{8} T^{8} \)
19$D_{4}$ \( ( 1 - T + 648 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 99 T + 5117 T^{2} + 183150 T^{3} + 4870902 T^{4} + 183150 p^{2} T^{5} + 5117 p^{4} T^{6} + 99 p^{6} T^{7} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 63 T + 2123 T^{2} - 50400 T^{3} + 1045362 T^{4} - 50400 p^{2} T^{5} + 2123 p^{4} T^{6} - 63 p^{6} T^{7} + p^{8} T^{8} \)
31$D_4\times C_2$ \( 1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4592 p^{2} T^{5} - 1217 p^{4} T^{6} + 7 p^{6} T^{7} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 + 32 T + 1806 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 18 T + 1913 T^{2} - 32490 T^{3} + 613812 T^{4} - 32490 p^{2} T^{5} + 1913 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \)
43$C_2^2$ \( ( 1 + 23 T - 1320 T^{2} + 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 81 T + 6929 T^{2} - 384102 T^{3} + 22437966 T^{4} - 384102 p^{2} T^{5} + 6929 p^{4} T^{6} - 81 p^{6} T^{7} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 7204 T^{2} + 26018214 T^{4} - 7204 p^{4} T^{6} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 + 126 T + 11993 T^{2} + 844326 T^{3} + 51207492 T^{4} + 844326 p^{2} T^{5} + 11993 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 10168 p^{2} T^{5} - 5513 p^{4} T^{6} - 41 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 - 116 T + 3787 T^{2} - 80156 T^{3} + 12934456 T^{4} - 80156 p^{2} T^{5} + 3787 p^{4} T^{6} - 116 p^{6} T^{7} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 18616 T^{2} + 137194926 T^{4} - 18616 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 43 T + 10452 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 83 T - 5459 T^{2} + 11122 T^{3} + 70528774 T^{4} + 11122 p^{2} T^{5} - 5459 p^{4} T^{6} - 83 p^{6} T^{7} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 81 T + 16289 T^{2} - 1142262 T^{3} + 166474326 T^{4} - 1142262 p^{2} T^{5} + 16289 p^{4} T^{6} - 81 p^{6} T^{7} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 6916 T^{2} + 69013446 T^{4} - 6916 p^{4} T^{6} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 + 196 T + 10291 T^{2} + 1824172 T^{3} + 341030200 T^{4} + 1824172 p^{2} T^{5} + 10291 p^{4} T^{6} + 196 p^{6} T^{7} + p^{8} T^{8} \)
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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

−10.03214952058346410208112793637, −9.457149691829423465927597262690, −9.303799228602202580264676005988, −9.088044384700959978361099416781, −8.489384269540464530431335121487, −8.362972879649692017780694826245, −8.305428035199002488750902065644, −7.74536939872732157704263464981, −7.74298474067207329194164358022, −7.14302323102791064878180342171, −6.78668673138639759254996227644, −6.55764872325180512179548165730, −6.39697447330836292472837099646, −6.02350480156748000235432463639, −5.65726420548311825099845956703, −5.15105172834436776771352073313, −4.53918976005238821994625486528, −4.15995692806504865553394132840, −3.90061714212670349806568205242, −3.86893486543229514033237225567, −3.57298324427366690752001843619, −2.81150799542191288782567888909, −1.95745800377030199383279169492, −1.51334409011988312388596239121, −0.54625460819383195273545340598, 0.54625460819383195273545340598, 1.51334409011988312388596239121, 1.95745800377030199383279169492, 2.81150799542191288782567888909, 3.57298324427366690752001843619, 3.86893486543229514033237225567, 3.90061714212670349806568205242, 4.15995692806504865553394132840, 4.53918976005238821994625486528, 5.15105172834436776771352073313, 5.65726420548311825099845956703, 6.02350480156748000235432463639, 6.39697447330836292472837099646, 6.55764872325180512179548165730, 6.78668673138639759254996227644, 7.14302323102791064878180342171, 7.74298474067207329194164358022, 7.74536939872732157704263464981, 8.305428035199002488750902065644, 8.362972879649692017780694826245, 8.489384269540464530431335121487, 9.088044384700959978361099416781, 9.303799228602202580264676005988, 9.457149691829423465927597262690, 10.03214952058346410208112793637

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