Properties

Label 8-384e4-1.1-c7e4-0-0
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 $0$

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: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(0.4423540912\)
\(L(\frac12)\) \(\approx\) \(0.4423540912\)
\(L(\frac{9}{2})\) 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

−6.92066291296960071338406357552, −6.44866303474674953400444187678, −6.36528240750520531786090191091, −6.16436844810119377021796029183, −5.95369404784908382296830774469, −5.84031228716819607963679042814, −5.36712953585746690102311733656, −5.30204884396201608139563361442, −5.19072240176508322155302361162, −4.84059617212196700263392833672, −4.42407594599404673328467001412, −4.30668628518889006039897038291, −4.17988318548107513241033593470, −3.43313404115976588345309133999, −3.22780206178922973234187744977, −3.13067269673774481599959378820, −2.82648197022283686964580929431, −1.97171917795230638677856483641, −1.76944423276890519388353042696, −1.74962375568427336776693155228, −1.61225868206442429239040177475, −1.01271362503436043365370149704, −0.69366825332640225199496103101, −0.68859000926799598859605948952, −0.081814977589238039949477988126, 0.081814977589238039949477988126, 0.68859000926799598859605948952, 0.69366825332640225199496103101, 1.01271362503436043365370149704, 1.61225868206442429239040177475, 1.74962375568427336776693155228, 1.76944423276890519388353042696, 1.97171917795230638677856483641, 2.82648197022283686964580929431, 3.13067269673774481599959378820, 3.22780206178922973234187744977, 3.43313404115976588345309133999, 4.17988318548107513241033593470, 4.30668628518889006039897038291, 4.42407594599404673328467001412, 4.84059617212196700263392833672, 5.19072240176508322155302361162, 5.30204884396201608139563361442, 5.36712953585746690102311733656, 5.84031228716819607963679042814, 5.95369404784908382296830774469, 6.16436844810119377021796029183, 6.36528240750520531786090191091, 6.44866303474674953400444187678, 6.92066291296960071338406357552

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