Properties

Label 8-384e4-1.1-c7e4-0-6
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $2.07055\times 10^{8}$
Root an. cond. $10.9524$
Motivic weight $7$
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  − 108·3-s − 336·5-s − 680·7-s + 7.29e3·9-s − 3.85e3·11-s − 1.06e4·13-s + 3.62e4·15-s + 2.62e4·17-s + 1.54e4·19-s + 7.34e4·21-s + 1.13e4·23-s − 2.02e4·25-s − 3.93e5·27-s + 1.85e3·29-s − 7.17e4·31-s + 4.16e5·33-s + 2.28e5·35-s + 1.80e5·37-s + 1.15e6·39-s + 1.12e4·41-s − 6.66e4·43-s − 2.44e6·45-s + 1.33e6·47-s − 2.15e5·49-s − 2.83e6·51-s + 8.64e5·53-s + 1.29e6·55-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.20·5-s − 0.749·7-s + 10/3·9-s − 0.873·11-s − 1.34·13-s + 2.77·15-s + 1.29·17-s + 0.516·19-s + 1.73·21-s + 0.193·23-s − 0.259·25-s − 3.84·27-s + 0.0141·29-s − 0.432·31-s + 2.01·33-s + 0.900·35-s + 0.584·37-s + 3.11·39-s + 0.0254·41-s − 0.127·43-s − 4.00·45-s + 1.87·47-s − 0.261·49-s − 2.99·51-s + 0.797·53-s + 1.05·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2.07055\times 10^{8}\)
Root analytic conductor: \(10.9524\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 + p^{3} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 336 T + 133172 T^{2} + 9135472 p T^{3} + 590476518 p^{2} T^{4} + 9135472 p^{8} T^{5} + 133172 p^{14} T^{6} + 336 p^{21} T^{7} + p^{28} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 680 T + 677716 T^{2} + 588392552 T^{3} + 939725142022 T^{4} + 588392552 p^{7} T^{5} + 677716 p^{14} T^{6} + 680 p^{21} T^{7} + p^{28} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 3856 T + 30530540 T^{2} + 132976308368 T^{3} + 1047263173715318 T^{4} + 132976308368 p^{7} T^{5} + 30530540 p^{14} T^{6} + 3856 p^{21} T^{7} + p^{28} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 10680 T + 120537388 T^{2} + 70615071304 p T^{3} + 6117210856743702 T^{4} + 70615071304 p^{8} T^{5} + 120537388 p^{14} T^{6} + 10680 p^{21} T^{7} + p^{28} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 26232 T + 1600605020 T^{2} - 31835394634952 T^{3} + 976739365216810374 T^{4} - 31835394634952 p^{7} T^{5} + 1600605020 p^{14} T^{6} - 26232 p^{21} T^{7} + p^{28} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 15456 T + 1923305452 T^{2} - 26004413281888 T^{3} + 1906547600676134550 T^{4} - 26004413281888 p^{7} T^{5} + 1923305452 p^{14} T^{6} - 15456 p^{21} T^{7} + p^{28} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 11312 T + 11969867132 T^{2} - 96044894964976 T^{3} + 58403450499455239142 T^{4} - 96044894964976 p^{7} T^{5} + 11969867132 p^{14} T^{6} - 11312 p^{21} T^{7} + p^{28} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 64 p T + 50403792500 T^{2} + 796188951014208 T^{3} + \)\(11\!\cdots\!38\)\( T^{4} + 796188951014208 p^{7} T^{5} + 50403792500 p^{14} T^{6} - 64 p^{22} T^{7} + p^{28} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 71752 T + 63197074228 T^{2} + 4508026689735624 T^{3} + \)\(22\!\cdots\!90\)\( T^{4} + 4508026689735624 p^{7} T^{5} + 63197074228 p^{14} T^{6} + 71752 p^{21} T^{7} + p^{28} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 180088 T + 12441033676 T^{2} - 6079371693226792 T^{3} - \)\(37\!\cdots\!82\)\( T^{4} - 6079371693226792 p^{7} T^{5} + 12441033676 p^{14} T^{6} - 180088 p^{21} T^{7} + p^{28} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 11224 T + 16688642524 p T^{2} - 12085470271439016 T^{3} + \)\(19\!\cdots\!02\)\( T^{4} - 12085470271439016 p^{7} T^{5} + 16688642524 p^{15} T^{6} - 11224 p^{21} T^{7} + p^{28} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 66688 T + 846851326732 T^{2} + 15686818765797504 T^{3} + \)\(31\!\cdots\!22\)\( T^{4} + 15686818765797504 p^{7} T^{5} + 846851326732 p^{14} T^{6} + 66688 p^{21} T^{7} + p^{28} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 1334448 T + 2187937371932 T^{2} - 1786090795383991792 T^{3} + \)\(16\!\cdots\!46\)\( T^{4} - 1786090795383991792 p^{7} T^{5} + 2187937371932 p^{14} T^{6} - 1334448 p^{21} T^{7} + p^{28} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 864576 T + 2840671248404 T^{2} - 2971751643465105600 T^{3} + \)\(39\!\cdots\!74\)\( T^{4} - 2971751643465105600 p^{7} T^{5} + 2840671248404 p^{14} T^{6} - 864576 p^{21} T^{7} + p^{28} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 1878448 T + 4950636262316 T^{2} + 8101870889030275888 T^{3} + \)\(15\!\cdots\!82\)\( T^{4} + 8101870889030275888 p^{7} T^{5} + 4950636262316 p^{14} T^{6} + 1878448 p^{21} T^{7} + p^{28} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 1901176 T + 8781687004780 T^{2} - 16554952821397109864 T^{3} + \)\(35\!\cdots\!54\)\( T^{4} - 16554952821397109864 p^{7} T^{5} + 8781687004780 p^{14} T^{6} - 1901176 p^{21} T^{7} + p^{28} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 5505488 T + 25639237833868 T^{2} + 79055690097472625872 T^{3} + \)\(22\!\cdots\!14\)\( T^{4} + 79055690097472625872 p^{7} T^{5} + 25639237833868 p^{14} T^{6} + 5505488 p^{21} T^{7} + p^{28} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 967696 T + 16348842760124 T^{2} - 2932478048048208 p^{2} T^{3} + \)\(15\!\cdots\!98\)\( T^{4} - 2932478048048208 p^{9} T^{5} + 16348842760124 p^{14} T^{6} - 967696 p^{21} T^{7} + p^{28} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 3244760 T + 13758579298108 T^{2} - 46662038113814919848 T^{3} + \)\(27\!\cdots\!62\)\( T^{4} - 46662038113814919848 p^{7} T^{5} + 13758579298108 p^{14} T^{6} - 3244760 p^{21} T^{7} + p^{28} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 6471816 T + 81916388031988 T^{2} + \)\(33\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!34\)\( T^{4} + \)\(33\!\cdots\!00\)\( p^{7} T^{5} + 81916388031988 p^{14} T^{6} + 6471816 p^{21} T^{7} + p^{28} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 17019600 T + 207642767089868 T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + \)\(98\!\cdots\!62\)\( T^{4} + \)\(15\!\cdots\!64\)\( p^{7} T^{5} + 207642767089868 p^{14} T^{6} + 17019600 p^{21} T^{7} + p^{28} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 13559816 T + 211747409486012 T^{2} - \)\(17\!\cdots\!48\)\( T^{3} + \)\(16\!\cdots\!30\)\( p T^{4} - \)\(17\!\cdots\!48\)\( p^{7} T^{5} + 211747409486012 p^{14} T^{6} - 13559816 p^{21} T^{7} + p^{28} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 2180520 T + 154114603199452 T^{2} - \)\(67\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!14\)\( T^{4} - \)\(67\!\cdots\!80\)\( p^{7} T^{5} + 154114603199452 p^{14} T^{6} - 2180520 p^{21} T^{7} + p^{28} T^{8} \)
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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

−7.46689539323484444303699986544, −7.22561625774164455897081452991, −7.22258454108167752417626441249, −6.66668762941156703564232640908, −6.46949231726278051542160700658, −6.10846872415068534220540638425, −5.82410267677645538282178894810, −5.79164335492914534225963065891, −5.69290826725178158743338253640, −5.00320690622622850782710730856, −4.98195525416833624609925621768, −4.85665379137817844051874012285, −4.65630939902686458880605826480, −4.12346556035308299699493315686, −3.84860367344059459266888699940, −3.71796038479912250217904216695, −3.43407002524394278601197747353, −3.09961472329585966961616204715, −2.43519852905487579036964026250, −2.41500904425416251528800173719, −2.35920277407033940006579881019, −1.30064735392108020694171767243, −1.28499226627412912629034395036, −1.15993189585949725218243672121, −0.72149593167168264004996416136, 0, 0, 0, 0, 0.72149593167168264004996416136, 1.15993189585949725218243672121, 1.28499226627412912629034395036, 1.30064735392108020694171767243, 2.35920277407033940006579881019, 2.41500904425416251528800173719, 2.43519852905487579036964026250, 3.09961472329585966961616204715, 3.43407002524394278601197747353, 3.71796038479912250217904216695, 3.84860367344059459266888699940, 4.12346556035308299699493315686, 4.65630939902686458880605826480, 4.85665379137817844051874012285, 4.98195525416833624609925621768, 5.00320690622622850782710730856, 5.69290826725178158743338253640, 5.79164335492914534225963065891, 5.82410267677645538282178894810, 6.10846872415068534220540638425, 6.46949231726278051542160700658, 6.66668762941156703564232640908, 7.22258454108167752417626441249, 7.22561625774164455897081452991, 7.46689539323484444303699986544

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