↑

Properties

Label 8-1575e4-1.1-c3e4-0-1
Degree $8$
Conductor $6.154\times 10^{12}$
Sign $1$
Analytic cond. $7.45738\times 10^{7}$
Root an. cond. $9.63991$
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  + 6·2-s + 10·4-s + 28·7-s + 3·8-s − 57·11-s − 43·13-s + 168·14-s + 54·16-s + 99·17-s − 12·19-s − 342·22-s + 156·23-s − 258·26-s + 280·28-s − 378·29-s − 93·31-s + 240·32-s + 594·34-s − 81·37-s − 72·38-s + 465·41-s + 64·43-s − 570·44-s + 936·46-s + 744·47-s + 490·49-s − 430·52-s + ⋯
L(s)  = 1  + 2.12·2-s + 5/4·4-s + 1.51·7-s + 0.132·8-s − 1.56·11-s − 0.917·13-s + 3.20·14-s + 0.843·16-s + 1.41·17-s − 0.144·19-s − 3.31·22-s + 1.41·23-s − 1.94·26-s + 1.88·28-s − 2.42·29-s − 0.538·31-s + 1.32·32-s + 2.99·34-s − 0.359·37-s − 0.307·38-s + 1.77·41-s + 0.226·43-s − 1.95·44-s + 3.00·46-s + 2.30·47-s + 10/7·49-s − 1.14·52-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(7.45738\times 10^{7}\)
Root analytic conductor: \(9.63991\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(20.28982748\)
\(L(\frac12)\) \(\approx\) \(20.28982748\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - p T )^{4} \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - 3 p T + 13 p T^{2} - 99 T^{3} + 149 p T^{4} - 99 p^{3} T^{5} + 13 p^{7} T^{6} - 3 p^{10} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 57 T + 4001 T^{2} + 182772 T^{3} + 7206874 T^{4} + 182772 p^{3} T^{5} + 4001 p^{6} T^{6} + 57 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 43 T + 4464 T^{2} + 221093 T^{3} + 13201958 T^{4} + 221093 p^{3} T^{5} + 4464 p^{6} T^{6} + 43 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 99 T + 13790 T^{2} - 926229 T^{3} + 90687946 T^{4} - 926229 p^{3} T^{5} + 13790 p^{6} T^{6} - 99 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 12 T - 10212 T^{2} + 154020 T^{3} + 95206646 T^{4} + 154020 p^{3} T^{5} - 10212 p^{6} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 156 T + 28082 T^{2} - 3613896 T^{3} + 479553835 T^{4} - 3613896 p^{3} T^{5} + 28082 p^{6} T^{6} - 156 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 378 T + 143760 T^{2} + 1025172 p T^{3} + 5852097929 T^{4} + 1025172 p^{4} T^{5} + 143760 p^{6} T^{6} + 378 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 3 p T + 16330 T^{2} - 936891 T^{3} + 385099458 T^{4} - 936891 p^{3} T^{5} + 16330 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 81 T + 107185 T^{2} + 469746 T^{3} + 5937897606 T^{4} + 469746 p^{3} T^{5} + 107185 p^{6} T^{6} + 81 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 465 T + 105716 T^{2} - 566235 T^{3} - 2050285754 T^{4} - 566235 p^{3} T^{5} + 105716 p^{6} T^{6} - 465 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 64 T + 64662 T^{2} + 2492176 T^{3} + 5477627255 T^{4} + 2492176 p^{3} T^{5} + 64662 p^{6} T^{6} - 64 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 744 T + 588276 T^{2} - 247094976 T^{3} + 101020317878 T^{4} - 247094976 p^{3} T^{5} + 588276 p^{6} T^{6} - 744 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 729 T + 333446 T^{2} - 137056347 T^{3} + 63585804370 T^{4} - 137056347 p^{3} T^{5} + 333446 p^{6} T^{6} - 729 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 231 T + 701280 T^{2} + 148812447 T^{3} + 204019205798 T^{4} + 148812447 p^{3} T^{5} + 701280 p^{6} T^{6} + 231 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 1353 T + 1374808 T^{2} + 931878711 T^{3} + 511309944510 T^{4} + 931878711 p^{3} T^{5} + 1374808 p^{6} T^{6} + 1353 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 1487 T + 1603371 T^{2} + 1080212116 T^{3} + 674138302124 T^{4} + 1080212116 p^{3} T^{5} + 1603371 p^{6} T^{6} + 1487 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 1725 T + 1434819 T^{2} - 706812600 T^{3} + 347044577276 T^{4} - 706812600 p^{3} T^{5} + 1434819 p^{6} T^{6} - 1725 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 512 T + 1030980 T^{2} + 484092568 T^{3} + 566560915526 T^{4} + 484092568 p^{3} T^{5} + 1030980 p^{6} T^{6} + 512 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 1629 T + 1989837 T^{2} - 1837769472 T^{3} + 1504809477470 T^{4} - 1837769472 p^{3} T^{5} + 1989837 p^{6} T^{6} - 1629 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 321 T + 661434 T^{2} - 144132717 T^{3} + 79870144034 T^{4} - 144132717 p^{3} T^{5} + 661434 p^{6} T^{6} - 321 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 978 T + 526228 T^{2} + 159100050 T^{3} - 360762420714 T^{4} + 159100050 p^{3} T^{5} + 526228 p^{6} T^{6} - 978 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 2616 T + 5183100 T^{2} + 6945540936 T^{3} + 7789652954246 T^{4} + 6945540936 p^{3} T^{5} + 5183100 p^{6} T^{6} + 2616 p^{9} T^{7} + p^{12} T^{8} \)
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

−6.11022503844699043753583525490, −5.81528602504639125278818984591, −5.72785911448922691678448777804, −5.54710812613582622688643754773, −5.42356670990799494835259755944, −5.06317534180188349411491486236, −4.97860836032626804527140371434, −4.87150717242869349285469048366, −4.82110425338040218458222715835, −4.19483807229602355175289279528, −4.10309909681837281106141139984, −4.06978432001380651431915621168, −3.98027756180955965345425804190, −3.25079084071184732101874032842, −3.24223155243258649142400444230, −3.18299664900038292049732060009, −2.65449876408670224317334151952, −2.33856158935470784456520370408, −2.28679113450539817336637444312, −1.82977136159468825754415174508, −1.66133540625240165111280020967, −1.26658701767835621029768382095, −0.74670882649796384997303032885, −0.70819486668873435668825550316, −0.30860168497251271866437364121, 0.30860168497251271866437364121, 0.70819486668873435668825550316, 0.74670882649796384997303032885, 1.26658701767835621029768382095, 1.66133540625240165111280020967, 1.82977136159468825754415174508, 2.28679113450539817336637444312, 2.33856158935470784456520370408, 2.65449876408670224317334151952, 3.18299664900038292049732060009, 3.24223155243258649142400444230, 3.25079084071184732101874032842, 3.98027756180955965345425804190, 4.06978432001380651431915621168, 4.10309909681837281106141139984, 4.19483807229602355175289279528, 4.82110425338040218458222715835, 4.87150717242869349285469048366, 4.97860836032626804527140371434, 5.06317534180188349411491486236, 5.42356670990799494835259755944, 5.54710812613582622688643754773, 5.72785911448922691678448777804, 5.81528602504639125278818984591, 6.11022503844699043753583525490

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