↑

Properties

Label 8-1470e4-1.1-c3e4-0-3
Degree $8$
Conductor $4.669\times 10^{12}$
Sign $1$
Analytic cond. $5.65892\times 10^{7}$
Root an. cond. $9.31304$
Motivic weight $3$
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  + 8·2-s + 12·3-s + 40·4-s − 20·5-s + 96·6-s + 160·8-s + 90·9-s − 160·10-s + 60·11-s + 480·12-s − 28·13-s − 240·15-s + 560·16-s + 56·17-s + 720·18-s + 28·19-s − 800·20-s + 480·22-s + 240·23-s + 1.92e3·24-s + 250·25-s − 224·26-s + 540·27-s + 252·29-s − 1.92e3·30-s − 200·31-s + 1.79e3·32-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s + 7.07·8-s + 10/3·9-s − 5.05·10-s + 1.64·11-s + 11.5·12-s − 0.597·13-s − 4.13·15-s + 35/4·16-s + 0.798·17-s + 9.42·18-s + 0.338·19-s − 8.94·20-s + 4.65·22-s + 2.17·23-s + 16.3·24-s + 2·25-s − 1.68·26-s + 3.84·27-s + 1.61·29-s − 11.6·30-s − 1.15·31-s + 9.89·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(5.65892\times 10^{7}\)
Root analytic conductor: \(9.31304\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(347.6528110\)
\(L(\frac12)\) \(\approx\) \(347.6528110\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
3$C_1$ \( ( 1 - p T )^{4} \)
5$C_1$ \( ( 1 + p T )^{4} \)
7 \( 1 \)
good11$C_2 \wr C_2\wr C_2$ \( 1 - 60 T + 2606 T^{2} - 144396 T^{3} + 6268642 T^{4} - 144396 p^{3} T^{5} + 2606 p^{6} T^{6} - 60 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 28 T + 912 T^{2} + 29988 T^{3} + 7688110 T^{4} + 29988 p^{3} T^{5} + 912 p^{6} T^{6} + 28 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 56 T + 11810 T^{2} - 526376 T^{3} + 68822594 T^{4} - 526376 p^{3} T^{5} + 11810 p^{6} T^{6} - 56 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 28 T + 21142 T^{2} - 409612 T^{3} + 195288962 T^{4} - 409612 p^{3} T^{5} + 21142 p^{6} T^{6} - 28 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 240 T + 59066 T^{2} - 7739688 T^{3} + 1076440258 T^{4} - 7739688 p^{3} T^{5} + 59066 p^{6} T^{6} - 240 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 252 T + 102742 T^{2} - 16911580 T^{3} + 3780510178 T^{4} - 16911580 p^{3} T^{5} + 102742 p^{6} T^{6} - 252 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 200 T + 64288 T^{2} + 9534968 T^{3} + 2556439778 T^{4} + 9534968 p^{3} T^{5} + 64288 p^{6} T^{6} + 200 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 284 T + 92686 T^{2} - 10194524 T^{3} + 3755246882 T^{4} - 10194524 p^{3} T^{5} + 92686 p^{6} T^{6} - 284 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 104 T + 213644 T^{2} - 16996952 T^{3} + 20749772678 T^{4} - 16996952 p^{3} T^{5} + 213644 p^{6} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 636 T + 321606 T^{2} - 116205468 T^{3} + 36115261058 T^{4} - 116205468 p^{3} T^{5} + 321606 p^{6} T^{6} - 636 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 720 T + 506810 T^{2} - 220813560 T^{3} + 83623336258 T^{4} - 220813560 p^{3} T^{5} + 506810 p^{6} T^{6} - 720 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 844 T + 240774 T^{2} + 28823172 T^{3} - 33955895550 T^{4} + 28823172 p^{3} T^{5} + 240774 p^{6} T^{6} - 844 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 4 p T + 696880 T^{2} - 103391052 T^{3} + 198893675246 T^{4} - 103391052 p^{3} T^{5} + 696880 p^{6} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 764 T + 839878 T^{2} - 509319692 T^{3} + 278597360642 T^{4} - 509319692 p^{3} T^{5} + 839878 p^{6} T^{6} - 764 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 1188 T + 1195894 T^{2} - 864569268 T^{3} + 500840871490 T^{4} - 864569268 p^{3} T^{5} + 1195894 p^{6} T^{6} - 1188 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 1552 T + 2022060 T^{2} - 1568527056 T^{3} + 1125832855302 T^{4} - 1568527056 p^{3} T^{5} + 2022060 p^{6} T^{6} - 1552 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 32 T + 670740 T^{2} - 186513920 T^{3} + 268972566518 T^{4} - 186513920 p^{3} T^{5} + 670740 p^{6} T^{6} + 32 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 676 T + 1169464 T^{2} - 6370156 p T^{3} + 630552180302 T^{4} - 6370156 p^{4} T^{5} + 1169464 p^{6} T^{6} - 676 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 276 T + 2135584 T^{2} + 470825012 T^{3} + 1788836409070 T^{4} + 470825012 p^{3} T^{5} + 2135584 p^{6} T^{6} + 276 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 664 T + 2385324 T^{2} - 1108656552 T^{3} + 2342667226246 T^{4} - 1108656552 p^{3} T^{5} + 2385324 p^{6} T^{6} - 664 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 552 T + 3585524 T^{2} + 1444141944 T^{3} + 4871994895574 T^{4} + 1444141944 p^{3} T^{5} + 3585524 p^{6} T^{6} + 552 p^{9} T^{7} + p^{12} T^{8} \)
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

−6.53988808911895771087878008439, −6.04968190527528779398226647913, −5.84883027642062040361740419236, −5.76515310609216101587935808126, −5.37310242393689089260197846399, −5.17115788285777556365077070852, −4.88610363606535460634954118562, −4.64376328775395496920705650086, −4.56875121132771340386011041634, −4.10645659723435367772724615694, −4.10357476677683488865415758963, −3.91220587257626619264739257554, −3.85781146156806216136474370660, −3.26407173334667772732081000250, −3.22285057132597004018613188544, −3.17207043311780279733676884994, −3.07996725747570895664977221881, −2.33767252836393246835234325383, −2.19387625927038971261414054127, −2.17775945389255329173253160380, −2.13724341784527653481517300746, −0.966660194586332523894573337267, −0.944977498331936044041302777308, −0.915837829686468822625915492419, −0.843368968648182063515846256574, 0.843368968648182063515846256574, 0.915837829686468822625915492419, 0.944977498331936044041302777308, 0.966660194586332523894573337267, 2.13724341784527653481517300746, 2.17775945389255329173253160380, 2.19387625927038971261414054127, 2.33767252836393246835234325383, 3.07996725747570895664977221881, 3.17207043311780279733676884994, 3.22285057132597004018613188544, 3.26407173334667772732081000250, 3.85781146156806216136474370660, 3.91220587257626619264739257554, 4.10357476677683488865415758963, 4.10645659723435367772724615694, 4.56875121132771340386011041634, 4.64376328775395496920705650086, 4.88610363606535460634954118562, 5.17115788285777556365077070852, 5.37310242393689089260197846399, 5.76515310609216101587935808126, 5.84883027642062040361740419236, 6.04968190527528779398226647913, 6.53988808911895771087878008439

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