Properties

Label 8-882e4-1.1-c2e4-0-6
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 12·5-s − 12·11-s + 12·17-s + 12·19-s + 24·20-s + 12·23-s + 40·25-s − 120·29-s − 72·31-s + 40·37-s − 128·43-s + 24·44-s + 168·47-s − 108·53-s + 144·55-s + 168·59-s − 24·61-s + 8·64-s − 88·67-s − 24·68-s + 120·71-s + 24·73-s − 24·76-s − 64·79-s − 144·85-s − 324·89-s + ⋯
L(s)  = 1  − 1/2·4-s − 2.39·5-s − 1.09·11-s + 0.705·17-s + 0.631·19-s + 6/5·20-s + 0.521·23-s + 8/5·25-s − 4.13·29-s − 2.32·31-s + 1.08·37-s − 2.97·43-s + 6/11·44-s + 3.57·47-s − 2.03·53-s + 2.61·55-s + 2.84·59-s − 0.393·61-s + 1/8·64-s − 1.31·67-s − 0.352·68-s + 1.69·71-s + 0.328·73-s − 0.315·76-s − 0.810·79-s − 1.69·85-s − 3.64·89-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3214522108\)
\(L(\frac12)\) \(\approx\) \(0.3214522108\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3 \( 1 \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 + 12 T + 104 T^{2} + 672 T^{3} + 3711 T^{4} + 672 p^{2} T^{5} + 104 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 + 12 T - 116 T^{2} + 216 T^{3} + 35535 T^{4} + 216 p^{2} T^{5} - 116 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
13$D_4\times C_2$ \( 1 - 244 T^{2} + 53574 T^{4} - 244 p^{4} T^{6} + p^{8} T^{8} \)
17$C_2^3$ \( 1 - 12 T - 88 T^{2} + 96 p T^{3} - 177 p^{2} T^{4} + 96 p^{3} T^{5} - 88 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 12 T + 398 T^{2} - 4200 T^{3} + 9507 T^{4} - 4200 p^{2} T^{5} + 398 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 12 T - 788 T^{2} + 1512 T^{3} + 512607 T^{4} + 1512 p^{2} T^{5} - 788 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \)
29$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{4} \)
31$D_4\times C_2$ \( 1 + 72 T + 3218 T^{2} + 107280 T^{3} + 2957187 T^{4} + 107280 p^{2} T^{5} + 3218 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 - 40 T + 1054 T^{2} + 87680 T^{3} - 3766445 T^{4} + 87680 p^{2} T^{5} + 1054 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 4960 T^{2} + 11110434 T^{4} - 4960 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 64 T + 2922 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 168 T + 16082 T^{2} - 23856 p T^{3} + 27363 p^{2} T^{4} - 23856 p^{3} T^{5} + 16082 p^{4} T^{6} - 168 p^{6} T^{7} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 + 108 T + 3418 T^{2} + 283824 T^{3} + 27341859 T^{4} + 283824 p^{2} T^{5} + 3418 p^{4} T^{6} + 108 p^{6} T^{7} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 168 T + 16322 T^{2} - 1161552 T^{3} + 68435283 T^{4} - 1161552 p^{2} T^{5} + 16322 p^{4} T^{6} - 168 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 + 24 T + 7586 T^{2} + 177456 T^{3} + 41539827 T^{4} + 177456 p^{2} T^{5} + 7586 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 + 88 T - 2882 T^{2} + 145024 T^{3} + 57997939 T^{4} + 145024 p^{2} T^{5} - 2882 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} \)
71$D_{4}$ \( ( 1 - 60 T + 4484 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 24 T + 6842 T^{2} - 159600 T^{3} + 16847427 T^{4} - 159600 p^{2} T^{5} + 6842 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \)
79$D_4\times C_2$ \( 1 + 64 T + 958 T^{2} - 598016 T^{3} - 54666173 T^{4} - 598016 p^{2} T^{5} + 958 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 16612 T^{2} + 134415078 T^{4} - 16612 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 + 324 T + 56936 T^{2} + 7109856 T^{3} + 695968527 T^{4} + 7109856 p^{2} T^{5} + 56936 p^{4} T^{6} + 324 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 24820 T^{2} + 317127462 T^{4} - 24820 p^{4} T^{6} + p^{8} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.19758335889708431287569027654, −6.91884221009537502781108332411, −6.82441761734371412336727588088, −6.60235926893671487906649371757, −5.85990298674309636420270965721, −5.71080775761467482705773313018, −5.56155294307697807657610488287, −5.48066156648677621799270363923, −5.42534375676695604602714255401, −4.99080483021600490046563195329, −4.44818494181542732715806952671, −4.42780002685877886887993365970, −4.13237682878854752850429058289, −3.86199587236489202408630599934, −3.82628814002723050118750941455, −3.34343254168462793801454324559, −3.32205760134875881783454661005, −2.98535274751071304275928654524, −2.67801784451326160488088142624, −1.99930189064364378008338899998, −1.84536494457328348153627609886, −1.67462447068852701387926046599, −0.865023435433351241376247024395, −0.45318610713704933851836536674, −0.18023651418260441292883668329, 0.18023651418260441292883668329, 0.45318610713704933851836536674, 0.865023435433351241376247024395, 1.67462447068852701387926046599, 1.84536494457328348153627609886, 1.99930189064364378008338899998, 2.67801784451326160488088142624, 2.98535274751071304275928654524, 3.32205760134875881783454661005, 3.34343254168462793801454324559, 3.82628814002723050118750941455, 3.86199587236489202408630599934, 4.13237682878854752850429058289, 4.42780002685877886887993365970, 4.44818494181542732715806952671, 4.99080483021600490046563195329, 5.42534375676695604602714255401, 5.48066156648677621799270363923, 5.56155294307697807657610488287, 5.71080775761467482705773313018, 5.85990298674309636420270965721, 6.60235926893671487906649371757, 6.82441761734371412336727588088, 6.91884221009537502781108332411, 7.19758335889708431287569027654

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