Properties

Label 8-1386e4-1.1-c1e4-0-1
Degree $8$
Conductor $3.690\times 10^{12}$
Sign $1$
Analytic cond. $15002.4$
Root an. cond. $3.32675$
Motivic weight $1$
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  + 2·2-s + 4-s + 2·5-s − 2·7-s − 2·8-s + 4·10-s + 2·11-s − 16·13-s − 4·14-s − 4·16-s − 10·17-s + 4·19-s + 2·20-s + 4·22-s + 2·23-s + 9·25-s − 32·26-s − 2·28-s − 4·31-s − 2·32-s − 20·34-s − 4·35-s − 8·37-s + 8·38-s − 4·40-s + 4·41-s + 2·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s − 0.707·8-s + 1.26·10-s + 0.603·11-s − 4.43·13-s − 1.06·14-s − 16-s − 2.42·17-s + 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.417·23-s + 9/5·25-s − 6.27·26-s − 0.377·28-s − 0.718·31-s − 0.353·32-s − 3.42·34-s − 0.676·35-s − 1.31·37-s + 1.29·38-s − 0.632·40-s + 0.624·41-s + 0.301·44-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(15002.4\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1386} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.336915992\)
\(L(\frac12)\) \(\approx\) \(1.336915992\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( ( 1 - T + T^{2} )^{2} \)
3 \( 1 \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2} \)
good5$D_4\times C_2$ \( 1 - 2 T - p T^{2} + 2 T^{3} + 36 T^{4} + 2 p T^{5} - p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$D_{4}$ \( ( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 10 T + 49 T^{2} + 10 p T^{3} + 36 p T^{4} + 10 p^{2} T^{5} + 49 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 4 T - 18 T^{2} + 16 T^{3} + 491 T^{4} + 16 p T^{5} - 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 2 T - 25 T^{2} + 34 T^{3} + 220 T^{4} + 34 p T^{5} - 25 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_4\times C_2$ \( 1 + 4 T + 22 T^{2} - 272 T^{3} - 1421 T^{4} - 272 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 8 T + 6 T^{2} - 128 T^{3} - 373 T^{4} - 128 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 10 T - T^{2} + 70 T^{3} + 3292 T^{4} + 70 p T^{5} - p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 16 T + 94 T^{2} + 896 T^{3} + 9867 T^{4} + 896 p T^{5} + 94 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 4 T - 74 T^{2} + 112 T^{3} + 3675 T^{4} + 112 p T^{5} - 74 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 6 T - 93 T^{2} - 42 T^{3} + 10724 T^{4} - 42 p T^{5} - 93 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 10 T + 13 T^{2} + 470 T^{3} - 3620 T^{4} + 470 p T^{5} + 13 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 4 T - 126 T^{2} + 16 T^{3} + 13667 T^{4} + 16 p T^{5} - 126 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 18 T + 103 T^{2} + 1134 T^{3} + 17004 T^{4} + 1134 p T^{5} + 103 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 14 T + 207 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 8 T - 58 T^{2} + 448 T^{3} + 1267 T^{4} + 448 p T^{5} - 58 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 26 T + 355 T^{2} + 26 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.72154936986498187605117110998, −6.72077057347371523663387637081, −6.56569202647164996088199418031, −6.22618291882989683056977629327, −5.81244741522650276625272603062, −5.49057891625365421758971751259, −5.47412752164126205848966644415, −5.28979939127568708750041287234, −4.92873320033882545165258287102, −4.91916754104161788336795775564, −4.61817547246575413571789199545, −4.49925639639893120650247363208, −4.27276199434790416563689152513, −4.00041023137910432940469268598, −3.51848597231228586258581817514, −3.44163445302693279683894351679, −2.95874300717292081382872984800, −2.88633330558794700538444226758, −2.61357252844115598472692173870, −2.30229617999755516831390922854, −2.24456830159863625944498970796, −1.72812060966195884419057826533, −1.51528882943935289287348409276, −0.62620629884395417407779320090, −0.22659812491714184598963774322, 0.22659812491714184598963774322, 0.62620629884395417407779320090, 1.51528882943935289287348409276, 1.72812060966195884419057826533, 2.24456830159863625944498970796, 2.30229617999755516831390922854, 2.61357252844115598472692173870, 2.88633330558794700538444226758, 2.95874300717292081382872984800, 3.44163445302693279683894351679, 3.51848597231228586258581817514, 4.00041023137910432940469268598, 4.27276199434790416563689152513, 4.49925639639893120650247363208, 4.61817547246575413571789199545, 4.91916754104161788336795775564, 4.92873320033882545165258287102, 5.28979939127568708750041287234, 5.47412752164126205848966644415, 5.49057891625365421758971751259, 5.81244741522650276625272603062, 6.22618291882989683056977629327, 6.56569202647164996088199418031, 6.72077057347371523663387637081, 6.72154936986498187605117110998

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