Properties

Label 8-384e4-1.1-c9e4-0-7
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

−7.53017513445126080781514259116, −7.02507028302346620054186361026, −6.73490738941110028317069328441, −6.50768238979770726416168102148, −6.49666512362236833628825851884, −5.76249363315923724017950405391, −5.48738810857562782655614214705, −5.36807796612241096167530796992, −5.13377739859887749083933353244, −4.90570361634913085024799041745, −4.44586704483706981328820042181, −4.43126855045153843524805208425, −4.19115124437524872288212914506, −3.62188524565809946703357919507, −3.51047586940874864314524183230, −3.35487847191303872584025779352, −3.01560865572331279762419658956, −2.47280148681012415590651450253, −2.44183316397950868901030853809, −2.36809578734458942170170474728, −2.33906356220073651852521195366, −1.58012040536872639645512883543, −1.24901169578581814012648795590, −1.24081591209974937618002944600, −1.19340712840394425275024081767, 0, 0, 0, 0, 1.19340712840394425275024081767, 1.24081591209974937618002944600, 1.24901169578581814012648795590, 1.58012040536872639645512883543, 2.33906356220073651852521195366, 2.36809578734458942170170474728, 2.44183316397950868901030853809, 2.47280148681012415590651450253, 3.01560865572331279762419658956, 3.35487847191303872584025779352, 3.51047586940874864314524183230, 3.62188524565809946703357919507, 4.19115124437524872288212914506, 4.43126855045153843524805208425, 4.44586704483706981328820042181, 4.90570361634913085024799041745, 5.13377739859887749083933353244, 5.36807796612241096167530796992, 5.48738810857562782655614214705, 5.76249363315923724017950405391, 6.49666512362236833628825851884, 6.50768238979770726416168102148, 6.73490738941110028317069328441, 7.02507028302346620054186361026, 7.53017513445126080781514259116

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