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Properties

Label 8-1815e4-1.1-c1e4-0-7
Degree $8$
Conductor $1.085\times 10^{13}$
Sign $1$
Analytic cond. $44117.9$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4·3-s + 4-s + 4·5-s + 12·6-s + 6·7-s − 7·8-s + 10·9-s + 12·10-s + 4·12-s + 7·13-s + 18·14-s + 16·15-s − 10·16-s + 10·17-s + 30·18-s + 9·19-s + 4·20-s + 24·21-s − 3·23-s − 28·24-s + 10·25-s + 21·26-s + 20·27-s + 6·28-s + 15·29-s + 48·30-s + ⋯
L(s)  = 1  + 2.12·2-s + 2.30·3-s + 1/2·4-s + 1.78·5-s + 4.89·6-s + 2.26·7-s − 2.47·8-s + 10/3·9-s + 3.79·10-s + 1.15·12-s + 1.94·13-s + 4.81·14-s + 4.13·15-s − 5/2·16-s + 2.42·17-s + 7.07·18-s + 2.06·19-s + 0.894·20-s + 5.23·21-s − 0.625·23-s − 5.71·24-s + 2·25-s + 4.11·26-s + 3.84·27-s + 1.13·28-s + 2.78·29-s + 8.76·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(44117.9\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(91.97529322\)
\(L(\frac12)\) \(\approx\) \(91.97529322\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{4} \)
5$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + p^{3} T^{2} - 7 p T^{3} + 23 T^{4} - 7 p^{2} T^{5} + p^{5} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 38 T^{2} - 129 T^{3} + 433 T^{4} - 129 p T^{5} + 38 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 62 T^{2} - 267 T^{3} + 1263 T^{4} - 267 p T^{5} + 62 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 84 T^{2} - 500 T^{3} + 2277 T^{4} - 500 p T^{5} + 84 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 98 T^{2} - 529 T^{3} + 3003 T^{4} - 529 p T^{5} + 98 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 37 T^{2} + 78 T^{3} + 1093 T^{4} + 78 p T^{5} + 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 120 T^{2} - 895 T^{3} + 5777 T^{4} - 895 p T^{5} + 120 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 107 T^{2} + 602 T^{3} + 3463 T^{4} + 602 p T^{5} + 107 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 27 T^{2} + 120 T^{3} + 2801 T^{4} + 120 p T^{5} + 27 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 302 T^{2} - 2913 T^{3} + 523 p T^{4} - 2913 p T^{5} + 302 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 62 T^{2} - 375 T^{3} + 1909 T^{4} - 375 p T^{5} + 62 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 96 T^{2} + 7 p T^{3} + 4637 T^{4} + 7 p^{2} T^{5} + 96 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 128 T^{2} - 775 T^{3} + 7309 T^{4} - 775 p T^{5} + 128 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 21 T + 353 T^{2} + 3746 T^{3} + 34103 T^{4} + 3746 p T^{5} + 353 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 156 T^{2} - 1147 T^{3} + 12741 T^{4} - 1147 p T^{5} + 156 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - T + 168 T^{2} - 89 T^{3} + 14153 T^{4} - 89 p T^{5} + 168 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 227 T^{2} + 2382 T^{3} + 22433 T^{4} + 2382 p T^{5} + 227 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - T + 218 T^{2} - 135 T^{3} + 22101 T^{4} - 135 p T^{5} + 218 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 227 T^{2} + 662 T^{3} + 23215 T^{4} + 662 p T^{5} + 227 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 255 T^{2} - 600 T^{3} + 28593 T^{4} - 600 p T^{5} + 255 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 3 p T^{2} + 2360 T^{3} + 31893 T^{4} + 2360 p T^{5} + 3 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 476 T^{2} + 6179 T^{3} + 72367 T^{4} + 6179 p T^{5} + 476 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.57535766079166193415333779326, −5.99746669749041330780370836729, −5.93871752012208241956761582382, −5.92319534438457257042271176238, −5.72218829342170620948205678535, −5.32557192309145897798087092838, −5.19242919359963929494156329849, −5.04677735323847615249620167287, −4.94081810450433488104508055153, −4.47558276610164603896235502731, −4.37624537970547017974853502395, −4.29708468007252467834816122376, −4.09765111183345372406199364681, −3.49721483447658311495107856570, −3.40930732280279547373074580312, −3.37459815923837553501007013338, −3.28438806573780059091644503585, −2.77833282591387082970962317041, −2.59950914380238059770492480143, −2.24126647343059215216201602413, −1.92545257066914544775461073227, −1.57109557227935181812235434246, −1.29220227461572524215152317490, −1.10081133327573062986884208839, −1.02978008896587462937853714286, 1.02978008896587462937853714286, 1.10081133327573062986884208839, 1.29220227461572524215152317490, 1.57109557227935181812235434246, 1.92545257066914544775461073227, 2.24126647343059215216201602413, 2.59950914380238059770492480143, 2.77833282591387082970962317041, 3.28438806573780059091644503585, 3.37459815923837553501007013338, 3.40930732280279547373074580312, 3.49721483447658311495107856570, 4.09765111183345372406199364681, 4.29708468007252467834816122376, 4.37624537970547017974853502395, 4.47558276610164603896235502731, 4.94081810450433488104508055153, 5.04677735323847615249620167287, 5.19242919359963929494156329849, 5.32557192309145897798087092838, 5.72218829342170620948205678535, 5.92319534438457257042271176238, 5.93871752012208241956761582382, 5.99746669749041330780370836729, 6.57535766079166193415333779326

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