Properties

Label 8-384e4-1.1-c9e4-0-6
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $1.52994\times 10^{9}$
Root an. cond. $14.0632$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 324·3-s + 240·5-s − 4.84e3·7-s + 6.56e4·9-s − 9.96e4·11-s + 6.08e4·13-s + 7.77e4·15-s − 4.34e5·17-s + 6.31e5·19-s − 1.56e6·21-s − 7.49e5·23-s − 8.81e5·25-s + 1.06e7·27-s + 7.90e6·29-s − 1.13e7·31-s − 3.22e7·33-s − 1.16e6·35-s + 1.35e7·37-s + 1.97e7·39-s − 1.88e7·41-s + 1.41e7·43-s + 1.57e7·45-s + 3.77e7·47-s − 6.42e7·49-s − 1.40e8·51-s − 1.15e8·53-s − 2.39e7·55-s + ⋯
L(s)  = 1  + 2.30·3-s + 0.171·5-s − 0.761·7-s + 10/3·9-s − 2.05·11-s + 0.590·13-s + 0.396·15-s − 1.26·17-s + 1.11·19-s − 1.75·21-s − 0.558·23-s − 0.451·25-s + 3.84·27-s + 2.07·29-s − 2.20·31-s − 4.73·33-s − 0.130·35-s + 1.19·37-s + 1.36·39-s − 1.04·41-s + 0.632·43-s + 0.572·45-s + 1.12·47-s − 1.59·49-s − 2.91·51-s − 2.00·53-s − 0.352·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.52994\times 10^{9}\)
Root analytic conductor: \(14.0632\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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$C_1$ \( ( 1 - p^{4} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 48 p T + 939284 T^{2} + 236450096 p T^{3} - 167112133242 p^{2} T^{4} + 236450096 p^{10} T^{5} + 939284 p^{18} T^{6} - 48 p^{28} T^{7} + p^{36} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 4840 T + 12530764 p T^{2} + 12761367208 p^{2} T^{3} + 11457097294378 p^{3} T^{4} + 12761367208 p^{11} T^{5} + 12530764 p^{19} T^{6} + 4840 p^{27} T^{7} + p^{36} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 99664 T + 10589400716 T^{2} + 699518969642576 T^{3} + 38813237732845803926 T^{4} + 699518969642576 p^{9} T^{5} + 10589400716 p^{18} T^{6} + 99664 p^{27} T^{7} + p^{36} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 360 p^{2} T + 24714128908 T^{2} - 695713226418488 T^{3} + \)\(29\!\cdots\!86\)\( T^{4} - 695713226418488 p^{9} T^{5} + 24714128908 p^{18} T^{6} - 360 p^{29} T^{7} + p^{36} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 434952 T + 394190258396 T^{2} + 122601307335261368 T^{3} + \)\(64\!\cdots\!26\)\( T^{4} + 122601307335261368 p^{9} T^{5} + 394190258396 p^{18} T^{6} + 434952 p^{27} T^{7} + p^{36} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 631776 T + 1164722951116 T^{2} - 471204029388957920 T^{3} + \)\(52\!\cdots\!86\)\( T^{4} - 471204029388957920 p^{9} T^{5} + 1164722951116 p^{18} T^{6} - 631776 p^{27} T^{7} + p^{36} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 749392 T + 3347296061756 T^{2} + 4837140337367041424 T^{3} + \)\(64\!\cdots\!18\)\( T^{4} + 4837140337367041424 p^{9} T^{5} + 3347296061756 p^{18} T^{6} + 749392 p^{27} T^{7} + p^{36} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 7908544 T + 58432091245844 T^{2} - \)\(22\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} - \)\(22\!\cdots\!08\)\( p^{9} T^{5} + 58432091245844 p^{18} T^{6} - 7908544 p^{27} T^{7} + p^{36} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 11351240 T + 115399090325812 T^{2} + \)\(71\!\cdots\!24\)\( T^{3} + \)\(43\!\cdots\!22\)\( T^{4} + \)\(71\!\cdots\!24\)\( p^{9} T^{5} + 115399090325812 p^{18} T^{6} + 11351240 p^{27} T^{7} + p^{36} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 13592920 T + 213368728836076 T^{2} + 87591136825729972088 T^{3} + \)\(27\!\cdots\!94\)\( T^{4} + 87591136825729972088 p^{9} T^{5} + 213368728836076 p^{18} T^{6} - 13592920 p^{27} T^{7} + p^{36} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 18838888 T + 656876228382716 T^{2} + \)\(15\!\cdots\!88\)\( T^{3} + \)\(73\!\cdots\!90\)\( p T^{4} + \)\(15\!\cdots\!88\)\( p^{9} T^{5} + 656876228382716 p^{18} T^{6} + 18838888 p^{27} T^{7} + p^{36} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 14177920 T + 1417437436848556 T^{2} - \)\(23\!\cdots\!96\)\( T^{3} + \)\(93\!\cdots\!54\)\( T^{4} - \)\(23\!\cdots\!96\)\( p^{9} T^{5} + 1417437436848556 p^{18} T^{6} - 14177920 p^{27} T^{7} + p^{36} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 37779120 T + 3062567184137372 T^{2} - \)\(92\!\cdots\!16\)\( T^{3} + \)\(49\!\cdots\!30\)\( T^{4} - \)\(92\!\cdots\!16\)\( p^{9} T^{5} + 3062567184137372 p^{18} T^{6} - 37779120 p^{27} T^{7} + p^{36} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 115336512 T + 10787477606282612 T^{2} + \)\(79\!\cdots\!64\)\( T^{3} + \)\(46\!\cdots\!26\)\( T^{4} + \)\(79\!\cdots\!64\)\( p^{9} T^{5} + 10787477606282612 p^{18} T^{6} + 115336512 p^{27} T^{7} + p^{36} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 115028080 T + 604137837099172 p T^{2} + \)\(28\!\cdots\!56\)\( T^{3} + \)\(46\!\cdots\!30\)\( T^{4} + \)\(28\!\cdots\!56\)\( p^{9} T^{5} + 604137837099172 p^{19} T^{6} + 115028080 p^{27} T^{7} + p^{36} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 173228648 T + 46925341200786892 T^{2} + \)\(58\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!42\)\( T^{4} + \)\(58\!\cdots\!20\)\( p^{9} T^{5} + 46925341200786892 p^{18} T^{6} + 173228648 p^{27} T^{7} + p^{36} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 231785104 T + 47923090341550828 T^{2} + \)\(16\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!82\)\( T^{4} + \)\(16\!\cdots\!56\)\( p^{9} T^{5} + 47923090341550828 p^{18} T^{6} + 231785104 p^{27} T^{7} + p^{36} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 197476208 T + 67580246245585148 T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!06\)\( T^{4} + \)\(16\!\cdots\!80\)\( p^{9} T^{5} + 67580246245585148 p^{18} T^{6} + 197476208 p^{27} T^{7} + p^{36} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 44629400 T + 155814395021474812 T^{2} - \)\(40\!\cdots\!56\)\( T^{3} + \)\(12\!\cdots\!82\)\( T^{4} - \)\(40\!\cdots\!56\)\( p^{9} T^{5} + 155814395021474812 p^{18} T^{6} - 44629400 p^{27} T^{7} + p^{36} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 355774584 T + 431793154567003252 T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!20\)\( p^{9} T^{5} + 431793154567003252 p^{18} T^{6} - 355774584 p^{27} T^{7} + p^{36} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 607613328 T + 689233408599137708 T^{2} + \)\(31\!\cdots\!68\)\( T^{3} + \)\(18\!\cdots\!34\)\( T^{4} + \)\(31\!\cdots\!68\)\( p^{9} T^{5} + 689233408599137708 p^{18} T^{6} + 607613328 p^{27} T^{7} + p^{36} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1157146424 T + 1248537877761010940 T^{2} + \)\(74\!\cdots\!24\)\( T^{3} + \)\(52\!\cdots\!78\)\( T^{4} + \)\(74\!\cdots\!24\)\( p^{9} T^{5} + 1248537877761010940 p^{18} T^{6} + 1157146424 p^{27} T^{7} + p^{36} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 1599536472 T + 3875436244314154204 T^{2} + \)\(37\!\cdots\!12\)\( T^{3} + \)\(47\!\cdots\!58\)\( T^{4} + \)\(37\!\cdots\!12\)\( p^{9} T^{5} + 3875436244314154204 p^{18} T^{6} + 1599536472 p^{27} T^{7} + p^{36} T^{8} \)
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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

−7.32341517323493663288230301025, −6.89013583105823440539572172014, −6.79613603927375148206941568988, −6.47616399761540225240966062116, −6.33931334871926930888267426712, −5.90061775399278663177776215348, −5.65596099447278706774452848843, −5.36652934096621254344459019422, −5.28086493928748137827446424779, −4.63939666340037243427760264116, −4.52488639727159675144653685501, −4.32359792631806797409138552493, −4.29177924002993912342583401000, −3.51242741038549639211717155367, −3.44086490343980813671160773258, −3.36568174501977654102308614959, −3.02559684708772189054496452466, −2.68146567459551211128253847097, −2.55854063715353987199450332367, −2.27870055679876075118294083488, −2.24973642552561119582822189716, −1.41744567510799243790706474929, −1.41154939171441550219092876756, −1.39843522108791600602660635531, −0.979350346058978359808742349398, 0, 0, 0, 0, 0.979350346058978359808742349398, 1.39843522108791600602660635531, 1.41154939171441550219092876756, 1.41744567510799243790706474929, 2.24973642552561119582822189716, 2.27870055679876075118294083488, 2.55854063715353987199450332367, 2.68146567459551211128253847097, 3.02559684708772189054496452466, 3.36568174501977654102308614959, 3.44086490343980813671160773258, 3.51242741038549639211717155367, 4.29177924002993912342583401000, 4.32359792631806797409138552493, 4.52488639727159675144653685501, 4.63939666340037243427760264116, 5.28086493928748137827446424779, 5.36652934096621254344459019422, 5.65596099447278706774452848843, 5.90061775399278663177776215348, 6.33931334871926930888267426712, 6.47616399761540225240966062116, 6.79613603927375148206941568988, 6.89013583105823440539572172014, 7.32341517323493663288230301025

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