Properties

Label 8-12e12-1.1-c1e4-0-19
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $36247.9$
Root an. cond. $3.71458$
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  − 2·5-s + 12·7-s + 4·11-s + 2·13-s − 16·17-s + 12·19-s + 12·23-s − 25-s + 6·29-s + 8·31-s − 24·35-s − 24·37-s − 18·41-s − 8·43-s + 8·47-s + 71·49-s − 16·53-s − 8·55-s + 12·59-s + 30·61-s − 4·65-s − 16·67-s + 48·77-s − 16·83-s + 32·85-s + 24·91-s − 24·95-s + ⋯
L(s)  = 1  − 0.894·5-s + 4.53·7-s + 1.20·11-s + 0.554·13-s − 3.88·17-s + 2.75·19-s + 2.50·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 4.05·35-s − 3.94·37-s − 2.81·41-s − 1.21·43-s + 1.16·47-s + 71/7·49-s − 2.19·53-s − 1.07·55-s + 1.56·59-s + 3.84·61-s − 0.496·65-s − 1.95·67-s + 5.47·77-s − 1.75·83-s + 3.47·85-s + 2.51·91-s − 2.46·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(36247.9\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.585339025\)
\(L(\frac12)\) \(\approx\) \(7.585339025\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 + 2 T + p T^{2} + 14 T^{3} + 16 T^{4} + 14 p T^{5} + p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 12 T + 73 T^{2} - 300 T^{3} + 912 T^{4} - 300 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 4 T + 5 T^{2} + 20 T^{3} - 164 T^{4} + 20 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$$\times$$C_2^2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 12 T + 97 T^{2} - 588 T^{3} + 2976 T^{4} - 588 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 6 T + 9 T^{2} + 6 p T^{3} - 1528 T^{4} + 6 p^{2} T^{5} + 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 8 T + 13 T^{2} + 88 T^{3} - 344 T^{4} + 88 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 2472 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + 18 T + 181 T^{2} + 1314 T^{3} + 8076 T^{4} + 1314 p T^{5} + 181 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 8 T + 65 T^{2} - 120 T^{3} - 604 T^{4} - 120 p T^{5} + 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 8 T - 19 T^{2} + 88 T^{3} + 1672 T^{4} + 88 p T^{5} - 19 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 16 T + 128 T^{2} + 1264 T^{3} + 11806 T^{4} + 1264 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 12 T + 45 T^{2} + 300 T^{3} - 5524 T^{4} + 300 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 30 T + 261 T^{2} + 702 T^{3} - 21664 T^{4} + 702 p T^{5} + 261 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 16 T + 65 T^{2} - 960 T^{3} - 14284 T^{4} - 960 p T^{5} + 65 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^3$ \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 16 T + 185 T^{2} + 784 T^{3} + 4900 T^{4} + 784 p T^{5} + 185 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 36 T^{2} + 13094 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.93699528084052113538421831170, −6.48205932740724701371288312357, −6.44365067813839869481846342219, −5.86897091429727433744391880252, −5.65066705617742639410644047192, −5.33450123377422134380977000547, −5.20025899053899096586262122925, −5.06295189108753553728248206334, −4.80039035018954299593411515833, −4.66450620206222159967428347305, −4.63541389630485940633144160847, −4.45214534279133566274377707576, −4.07274578880151473524991261477, −3.79488360529797966489264682391, −3.40975567995986542888361983456, −3.39546950288605650594126829191, −3.12247372255735707840465447485, −2.48331375142977001405731312398, −2.43156374927458278387823208067, −1.92178264714358623691457130842, −1.72350654814537397065349723163, −1.54016217719993292087881043581, −1.34394609449793411646463234882, −0.968189061979152230390646133909, −0.48408537918084996282703278621, 0.48408537918084996282703278621, 0.968189061979152230390646133909, 1.34394609449793411646463234882, 1.54016217719993292087881043581, 1.72350654814537397065349723163, 1.92178264714358623691457130842, 2.43156374927458278387823208067, 2.48331375142977001405731312398, 3.12247372255735707840465447485, 3.39546950288605650594126829191, 3.40975567995986542888361983456, 3.79488360529797966489264682391, 4.07274578880151473524991261477, 4.45214534279133566274377707576, 4.63541389630485940633144160847, 4.66450620206222159967428347305, 4.80039035018954299593411515833, 5.06295189108753553728248206334, 5.20025899053899096586262122925, 5.33450123377422134380977000547, 5.65066705617742639410644047192, 5.86897091429727433744391880252, 6.44365067813839869481846342219, 6.48205932740724701371288312357, 6.93699528084052113538421831170

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