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Properties

Label 8-2541e4-1.1-c1e4-0-1
Degree $8$
Conductor $4.169\times 10^{13}$
Sign $1$
Analytic cond. $169483.$
Root an. cond. $4.50444$
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 + 4·3-s − 2·5-s + 8·6-s − 4·7-s − 4·8-s + 10·9-s − 4·10-s + 10·13-s − 8·14-s − 8·15-s − 6·16-s + 6·17-s + 20·18-s + 18·19-s − 16·21-s − 2·23-s − 16·24-s − 12·25-s + 20·26-s + 20·27-s + 6·29-s − 16·30-s + 12·34-s + 8·35-s − 4·37-s + 36·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s − 0.894·5-s + 3.26·6-s − 1.51·7-s − 1.41·8-s + 10/3·9-s − 1.26·10-s + 2.77·13-s − 2.13·14-s − 2.06·15-s − 3/2·16-s + 1.45·17-s + 4.71·18-s + 4.12·19-s − 3.49·21-s − 0.417·23-s − 3.26·24-s − 2.39·25-s + 3.92·26-s + 3.84·27-s + 1.11·29-s − 2.92·30-s + 2.05·34-s + 1.35·35-s − 0.657·37-s + 5.83·38-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(169483.\)
Root analytic conductor: \(4.50444\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(27.03814448\)
\(L(\frac12)\) \(\approx\) \(27.03814448\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + p^{2} T^{2} - p^{2} T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 16 T^{2} + 28 T^{3} + 111 T^{4} + 28 p T^{5} + 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 84 T^{2} - 422 T^{3} + 1844 T^{4} - 422 p T^{5} + 84 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 28 T^{2} - 24 T^{3} + 111 T^{4} - 24 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 184 T^{2} - 1278 T^{3} + 6468 T^{4} - 1278 p T^{5} + 184 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 28 T^{2} + 190 T^{3} + 516 T^{4} + 190 p T^{5} + 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 20 T^{2} - 126 T^{3} + 1404 T^{4} - 126 p T^{5} + 20 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 28 T^{2} - 216 T^{3} + 606 T^{4} - 216 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 72 T^{2} + 284 T^{3} + 4082 T^{4} + 284 p T^{5} + 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 124 T^{2} + 48 T^{3} + 6810 T^{4} + 48 p T^{5} + 124 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 282 T^{2} - 2632 T^{3} + 19943 T^{4} - 2632 p T^{5} + 282 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 44 T^{2} + 504 T^{3} + 93 p T^{4} + 504 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 148 T^{2} - 192 T^{3} + 9942 T^{4} - 192 p T^{5} + 148 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 208 T^{2} + 936 T^{3} + 17751 T^{4} + 936 p T^{5} + 208 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 108 T^{2} + 314 T^{3} + 3596 T^{4} + 314 p T^{5} + 108 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 130 T^{2} + 56 T^{3} + 8299 T^{4} + 56 p T^{5} + 130 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 220 T^{2} + 1014 T^{3} + 20916 T^{4} + 1014 p T^{5} + 220 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 34 T + 708 T^{2} - 9614 T^{3} + 96764 T^{4} - 9614 p T^{5} + 708 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 280 T^{2} - 1584 T^{3} + 8754 T^{4} - 1584 p T^{5} + 280 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 260 T^{2} - 1188 T^{3} + 30039 T^{4} - 1188 p T^{5} + 260 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 352 T^{2} - 4068 T^{3} + 48255 T^{4} - 4068 p T^{5} + 352 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 4 p T^{2} + 2758 T^{3} + 56428 T^{4} + 2758 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.31040694133307296439764693047, −6.06049203599155211767209716435, −6.03070010377484614982457386292, −5.60519099053935560194585314743, −5.57857385569260262819008063124, −5.07591951749177202381743219898, −4.99305096851432735166275557852, −4.90594510692501495976375557107, −4.53956410924468408723668311126, −4.21318171166286894864072675440, −3.98429375328613011950614477817, −3.88282353355599189273972139150, −3.55970263067485521535417020492, −3.41234345253578277368079241114, −3.38515148867660655831840768202, −3.34674604456330356644430473746, −3.27696248668565379865974303419, −2.60073375527448046980541230878, −2.52309959858540580708823373241, −2.29994351920035005096452895197, −1.73262067994493523701920987863, −1.53269902999755525942341424192, −1.00541820926269119317767085626, −0.76279440149351108306993123535, −0.68115865227956834303287076665, 0.68115865227956834303287076665, 0.76279440149351108306993123535, 1.00541820926269119317767085626, 1.53269902999755525942341424192, 1.73262067994493523701920987863, 2.29994351920035005096452895197, 2.52309959858540580708823373241, 2.60073375527448046980541230878, 3.27696248668565379865974303419, 3.34674604456330356644430473746, 3.38515148867660655831840768202, 3.41234345253578277368079241114, 3.55970263067485521535417020492, 3.88282353355599189273972139150, 3.98429375328613011950614477817, 4.21318171166286894864072675440, 4.53956410924468408723668311126, 4.90594510692501495976375557107, 4.99305096851432735166275557852, 5.07591951749177202381743219898, 5.57857385569260262819008063124, 5.60519099053935560194585314743, 6.03070010377484614982457386292, 6.06049203599155211767209716435, 6.31040694133307296439764693047

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