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Properties

Label 6-2e18-1.1-c13e3-0-0
Degree $6$
Conductor $262144$
Sign $-1$
Analytic cond. $323221.$
Root an. cond. $8.28418$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 520·3-s − 1.15e4·5-s − 9.63e4·7-s − 2.35e6·9-s + 1.92e6·11-s + 1.12e7·13-s + 6.02e6·15-s − 1.68e7·17-s + 7.46e7·19-s + 5.00e7·21-s − 4.86e8·23-s − 1.54e9·25-s + 2.58e9·27-s − 1.74e9·29-s − 6.47e9·31-s − 1.00e9·33-s + 1.11e9·35-s + 1.18e9·37-s − 5.82e9·39-s − 1.21e10·41-s + 1.51e11·43-s + 2.73e10·45-s − 2.73e11·47-s − 1.54e11·49-s + 8.78e9·51-s − 1.51e11·53-s − 2.23e10·55-s + ⋯
L(s)  = 1  − 0.411·3-s − 0.331·5-s − 0.309·7-s − 1.47·9-s + 0.327·11-s + 0.644·13-s + 0.136·15-s − 0.169·17-s + 0.364·19-s + 0.127·21-s − 0.685·23-s − 1.26·25-s + 1.28·27-s − 0.545·29-s − 1.30·31-s − 0.134·33-s + 0.102·35-s + 0.0762·37-s − 0.265·39-s − 0.400·41-s + 3.64·43-s + 0.490·45-s − 3.70·47-s − 1.59·49-s + 0.0699·51-s − 0.938·53-s − 0.108·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+13/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(262144\)    =    \(2^{18}\)
Sign: $-1$
Analytic conductor: \(323221.\)
Root analytic conductor: \(8.28418\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 262144,\ (\ :13/2, 13/2, 13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) 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 \( 1 \)
good3$S_4\times C_2$ \( 1 + 520 T + 875027 p T^{2} + 57392 p^{4} T^{3} + 875027 p^{14} T^{4} + 520 p^{26} T^{5} + p^{39} T^{6} \)
5$S_4\times C_2$ \( 1 + 11594 T + 2683067 p^{4} T^{2} + 161098161036 p^{3} T^{3} + 2683067 p^{17} T^{4} + 11594 p^{26} T^{5} + p^{39} T^{6} \)
7$S_4\times C_2$ \( 1 + 96304 T + 163328757173 T^{2} + 2237546422955104 p T^{3} + 163328757173 p^{13} T^{4} + 96304 p^{26} T^{5} + p^{39} T^{6} \)
11$S_4\times C_2$ \( 1 - 1923816 T + 5650770904563 p T^{2} - 1037069791322583152 p^{2} T^{3} + 5650770904563 p^{14} T^{4} - 1923816 p^{26} T^{5} + p^{39} T^{6} \)
13$S_4\times C_2$ \( 1 - 11211006 T + 223839612932259 T^{2} - \)\(48\!\cdots\!36\)\( T^{3} + 223839612932259 p^{13} T^{4} - 11211006 p^{26} T^{5} + p^{39} T^{6} \)
17$S_4\times C_2$ \( 1 + 16892538 T + 4257721413354111 T^{2} + \)\(79\!\cdots\!12\)\( T^{3} + 4257721413354111 p^{13} T^{4} + 16892538 p^{26} T^{5} + p^{39} T^{6} \)
19$S_4\times C_2$ \( 1 - 74665048 T + 6254814151747283 p T^{2} - \)\(60\!\cdots\!64\)\( T^{3} + 6254814151747283 p^{14} T^{4} - 74665048 p^{26} T^{5} + p^{39} T^{6} \)
23$S_4\times C_2$ \( 1 + 486642576 T + 873164834223143973 T^{2} + \)\(15\!\cdots\!00\)\( T^{3} + 873164834223143973 p^{13} T^{4} + 486642576 p^{26} T^{5} + p^{39} T^{6} \)
29$S_4\times C_2$ \( 1 + 1747468946 T + 17696611820353905107 T^{2} + \)\(38\!\cdots\!76\)\( T^{3} + 17696611820353905107 p^{13} T^{4} + 1747468946 p^{26} T^{5} + p^{39} T^{6} \)
31$S_4\times C_2$ \( 1 + 6471356864 T + 32239427727278410973 T^{2} + \)\(96\!\cdots\!48\)\( T^{3} + 32239427727278410973 p^{13} T^{4} + 6471356864 p^{26} T^{5} + p^{39} T^{6} \)
37$S_4\times C_2$ \( 1 - 1189828278 T + \)\(54\!\cdots\!91\)\( T^{2} + \)\(41\!\cdots\!68\)\( T^{3} + \)\(54\!\cdots\!91\)\( p^{13} T^{4} - 1189828278 p^{26} T^{5} + p^{39} T^{6} \)
41$S_4\times C_2$ \( 1 + 12181275970 T + \)\(21\!\cdots\!83\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(21\!\cdots\!83\)\( p^{13} T^{4} + 12181275970 p^{26} T^{5} + p^{39} T^{6} \)
43$S_4\times C_2$ \( 1 - 151151885096 T + \)\(12\!\cdots\!29\)\( T^{2} - \)\(64\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!29\)\( p^{13} T^{4} - 151151885096 p^{26} T^{5} + p^{39} T^{6} \)
47$S_4\times C_2$ \( 1 + 273552663072 T + \)\(39\!\cdots\!37\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!37\)\( p^{13} T^{4} + 273552663072 p^{26} T^{5} + p^{39} T^{6} \)
53$S_4\times C_2$ \( 1 + 151381970362 T + \)\(63\!\cdots\!19\)\( T^{2} + \)\(70\!\cdots\!52\)\( T^{3} + \)\(63\!\cdots\!19\)\( p^{13} T^{4} + 151381970362 p^{26} T^{5} + p^{39} T^{6} \)
59$S_4\times C_2$ \( 1 - 534333367560 T + \)\(18\!\cdots\!37\)\( T^{2} - \)\(38\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!37\)\( p^{13} T^{4} - 534333367560 p^{26} T^{5} + p^{39} T^{6} \)
61$S_4\times C_2$ \( 1 + 598414222482 T + \)\(31\!\cdots\!23\)\( T^{2} + \)\(92\!\cdots\!44\)\( T^{3} + \)\(31\!\cdots\!23\)\( p^{13} T^{4} + 598414222482 p^{26} T^{5} + p^{39} T^{6} \)
67$S_4\times C_2$ \( 1 + 703511405320 T + \)\(11\!\cdots\!69\)\( T^{2} + \)\(44\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!69\)\( p^{13} T^{4} + 703511405320 p^{26} T^{5} + p^{39} T^{6} \)
71$S_4\times C_2$ \( 1 - 1867178467152 T + \)\(44\!\cdots\!33\)\( T^{2} - \)\(44\!\cdots\!44\)\( T^{3} + \)\(44\!\cdots\!33\)\( p^{13} T^{4} - 1867178467152 p^{26} T^{5} + p^{39} T^{6} \)
73$S_4\times C_2$ \( 1 + 2138161799730 T + \)\(36\!\cdots\!99\)\( T^{2} + \)\(44\!\cdots\!80\)\( T^{3} + \)\(36\!\cdots\!99\)\( p^{13} T^{4} + 2138161799730 p^{26} T^{5} + p^{39} T^{6} \)
79$S_4\times C_2$ \( 1 - 6521880375328 T + \)\(23\!\cdots\!17\)\( T^{2} - \)\(58\!\cdots\!84\)\( T^{3} + \)\(23\!\cdots\!17\)\( p^{13} T^{4} - 6521880375328 p^{26} T^{5} + p^{39} T^{6} \)
83$S_4\times C_2$ \( 1 + 5725896739752 T + \)\(68\!\cdots\!65\)\( T^{2} - \)\(15\!\cdots\!68\)\( T^{3} + \)\(68\!\cdots\!65\)\( p^{13} T^{4} + 5725896739752 p^{26} T^{5} + p^{39} T^{6} \)
89$S_4\times C_2$ \( 1 + 11930629520834 T + \)\(11\!\cdots\!71\)\( T^{2} + \)\(58\!\cdots\!68\)\( T^{3} + \)\(11\!\cdots\!71\)\( p^{13} T^{4} + 11930629520834 p^{26} T^{5} + p^{39} T^{6} \)
97$S_4\times C_2$ \( 1 + 23076828823050 T + \)\(34\!\cdots\!31\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!31\)\( p^{13} T^{4} + 23076828823050 p^{26} T^{5} + p^{39} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.29040776469482136446919010183, −10.89883848137639333085586738208, −10.85218586807430996663508507130, −10.18846412008831987235156272719, −9.429690436984604863494117067081, −9.353869837484197935206074048628, −9.342073348530649592818822880129, −8.242451857908815078494973994746, −8.151426600271322415842842141643, −8.120974453861076148044463869222, −7.34330921209602774299269725788, −6.68623373159049732632397284609, −6.64912973560252055237166322984, −5.94891673066216216728928373565, −5.60621592737284145340448194394, −5.51716014710187703388050806101, −4.81976039327959333333805338150, −4.08709079843671270120052662134, −4.02161361488029432278053218568, −3.37011693505837269681433575251, −2.90542463227553563041175571529, −2.64435378845333726235298341832, −1.81182706847567728057670544807, −1.51187148574974827301760487102, −0.991706103654347894325383301260, 0, 0, 0, 0.991706103654347894325383301260, 1.51187148574974827301760487102, 1.81182706847567728057670544807, 2.64435378845333726235298341832, 2.90542463227553563041175571529, 3.37011693505837269681433575251, 4.02161361488029432278053218568, 4.08709079843671270120052662134, 4.81976039327959333333805338150, 5.51716014710187703388050806101, 5.60621592737284145340448194394, 5.94891673066216216728928373565, 6.64912973560252055237166322984, 6.68623373159049732632397284609, 7.34330921209602774299269725788, 8.120974453861076148044463869222, 8.151426600271322415842842141643, 8.242451857908815078494973994746, 9.342073348530649592818822880129, 9.353869837484197935206074048628, 9.429690436984604863494117067081, 10.18846412008831987235156272719, 10.85218586807430996663508507130, 10.89883848137639333085586738208, 11.29040776469482136446919010183

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