↑

Properties

Label 8-4998e4-1.1-c1e4-0-0
Degree $8$
Conductor $6.240\times 10^{14}$
Sign $1$
Analytic cond. $2.53684\times 10^{6}$
Root an. cond. $6.31737$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 4·3-s + 10·4-s + 2·5-s + 16·6-s − 20·8-s + 10·9-s − 8·10-s − 2·11-s − 40·12-s − 6·13-s − 8·15-s + 35·16-s + 4·17-s − 40·18-s − 8·19-s + 20·20-s + 8·22-s − 8·23-s + 80·24-s − 5·25-s + 24·26-s − 20·27-s + 32·30-s − 16·31-s − 56·32-s + 8·33-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s + 0.894·5-s + 6.53·6-s − 7.07·8-s + 10/3·9-s − 2.52·10-s − 0.603·11-s − 11.5·12-s − 1.66·13-s − 2.06·15-s + 35/4·16-s + 0.970·17-s − 9.42·18-s − 1.83·19-s + 4.47·20-s + 1.70·22-s − 1.66·23-s + 16.3·24-s − 25-s + 4.70·26-s − 3.84·27-s + 5.84·30-s − 2.87·31-s − 9.89·32-s + 1.39·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(2.53684\times 10^{6}\)
Root analytic conductor: \(6.31737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1815988022\)
\(L(\frac12)\) \(\approx\) \(0.1815988022\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
3$C_1$ \( ( 1 + T )^{4} \)
7 \( 1 \)
17$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 9 T^{2} - 18 T^{3} + 68 T^{4} - 18 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 23 T^{2} + 70 T^{3} + 292 T^{4} + 70 p T^{5} + 23 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 53 T^{2} + 210 T^{3} + 1048 T^{4} + 210 p T^{5} + 53 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 66 T^{2} + 400 T^{3} + 1794 T^{4} + 400 p T^{5} + 66 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 90 T^{2} + 496 T^{3} + 3138 T^{4} + 496 p T^{5} + 90 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 90 T^{2} + 16 T^{3} + 3650 T^{4} + 16 p T^{5} + 90 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 194 T^{2} + 1520 T^{3} + 9994 T^{4} + 1520 p T^{5} + 194 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 191 T^{2} - 1434 T^{3} + 10854 T^{4} - 1434 p T^{5} + 191 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 118 T^{2} - 236 T^{3} + 6058 T^{4} - 236 p T^{5} + 118 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 139 T^{2} + 818 T^{3} + 7504 T^{4} + 818 p T^{5} + 139 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 210 T^{2} - 1808 T^{3} + 14058 T^{4} - 1808 p T^{5} + 210 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 381 T^{2} - 4026 T^{3} + 35336 T^{4} - 4026 p T^{5} + 381 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 186 T^{2} - 80 T^{3} + 14954 T^{4} - 80 p T^{5} + 186 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 128 T^{2} - 1116 T^{3} + 7950 T^{4} - 1116 p T^{5} + 128 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 243 T^{2} - 2110 T^{3} + 23016 T^{4} - 2110 p T^{5} + 243 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 140 T^{2} + 876 T^{3} + 10466 T^{4} + 876 p T^{5} + 140 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 289 T^{2} + 2638 T^{3} + 31180 T^{4} + 2638 p T^{5} + 289 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 267 T^{2} + 1470 T^{3} + 29712 T^{4} + 1470 p T^{5} + 267 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 329 T^{2} + 1442 T^{3} + 40790 T^{4} + 1442 p T^{5} + 329 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 113 T^{2} + 18 T^{3} + 2004 T^{4} + 18 p T^{5} + 113 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 305 T^{2} + 554 T^{3} + 41724 T^{4} + 554 p T^{5} + 305 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.93222952270688700720373778152, −5.62142260604899270096603970472, −5.60055066507577730379037516535, −5.50091818742344070561212349531, −5.39567781131264396135413898990, −4.82574212925470092711166664960, −4.78027013519972050446328297860, −4.48965024087126798221992058552, −4.43399504520816701681877048999, −3.89750753571202855876029794849, −3.79284789426554225161974370036, −3.63905692881399393868838247817, −3.63668550924576568770405306609, −2.76580563325471965654633476100, −2.68507945156769077155433904614, −2.52461561645697836400680453530, −2.42994177457531334329852226076, −1.95493935914795495707181341693, −1.78873234375759093558377926975, −1.70998867100284142414638614380, −1.62529552843350456836571674612, −0.878390703355193312171328473342, −0.59588323212512492740129530060, −0.58922153763451302634658408400, −0.21177702193388315970378481770, 0.21177702193388315970378481770, 0.58922153763451302634658408400, 0.59588323212512492740129530060, 0.878390703355193312171328473342, 1.62529552843350456836571674612, 1.70998867100284142414638614380, 1.78873234375759093558377926975, 1.95493935914795495707181341693, 2.42994177457531334329852226076, 2.52461561645697836400680453530, 2.68507945156769077155433904614, 2.76580563325471965654633476100, 3.63668550924576568770405306609, 3.63905692881399393868838247817, 3.79284789426554225161974370036, 3.89750753571202855876029794849, 4.43399504520816701681877048999, 4.48965024087126798221992058552, 4.78027013519972050446328297860, 4.82574212925470092711166664960, 5.39567781131264396135413898990, 5.50091818742344070561212349531, 5.60055066507577730379037516535, 5.62142260604899270096603970472, 5.93222952270688700720373778152

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