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Properties

Label 6-2e18-1.1-c19e3-0-1
Degree $6$
Conductor $262144$
Sign $-1$
Analytic cond. $3.14053\times 10^{6}$
Root an. cond. $12.1013$
Motivic weight $19$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.37e4·3-s − 2.14e6·5-s − 5.58e7·7-s − 1.13e9·9-s − 2.97e8·11-s + 1.48e10·13-s − 5.07e10·15-s + 8.03e11·17-s + 3.21e12·19-s − 1.32e12·21-s − 2.49e13·23-s + 9.85e12·25-s − 1.93e13·27-s − 7.76e13·29-s + 2.48e14·31-s − 7.05e12·33-s + 1.19e14·35-s + 4.14e14·37-s + 3.52e14·39-s + 2.81e15·41-s − 6.66e15·43-s + 2.43e15·45-s − 1.50e15·47-s + 4.28e15·49-s + 1.90e16·51-s − 5.60e16·53-s + 6.36e14·55-s + ⋯
L(s)  = 1  + 0.696·3-s − 0.490·5-s − 0.523·7-s − 0.979·9-s − 0.0380·11-s + 0.388·13-s − 0.341·15-s + 1.64·17-s + 2.28·19-s − 0.364·21-s − 2.88·23-s + 0.516·25-s − 0.487·27-s − 0.994·29-s + 1.68·31-s − 0.0264·33-s + 0.256·35-s + 0.524·37-s + 0.270·39-s + 1.34·41-s − 2.02·43-s + 0.479·45-s − 0.195·47-s + 0.375·49-s + 1.14·51-s − 2.33·53-s + 0.0186·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+19/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(262144\)    =    \(2^{18}\)
Sign: $-1$
Analytic conductor: \(3.14053\times 10^{6}\)
Root analytic conductor: \(12.1013\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 262144,\ (\ :19/2, 19/2, 19/2),\ -1)\)

Particular Values

\(L(10)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{21}{2})\) 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$S_4\times C_2$ \( 1 - 23732 T + 63022931 p^{3} T^{2} - 65977129592 p^{6} T^{3} + 63022931 p^{22} T^{4} - 23732 p^{38} T^{5} + p^{57} T^{6} \)
5$S_4\times C_2$ \( 1 + 2140218 T - 1053936821721 p T^{2} - 124256106401873828 p^{4} T^{3} - 1053936821721 p^{20} T^{4} + 2140218 p^{38} T^{5} + p^{57} T^{6} \)
7$S_4\times C_2$ \( 1 + 55851720 T - 23724723206571 p^{2} T^{2} - \)\(49\!\cdots\!68\)\( p^{4} T^{3} - 23724723206571 p^{21} T^{4} + 55851720 p^{38} T^{5} + p^{57} T^{6} \)
11$S_4\times C_2$ \( 1 + 27035724 p T + 15323314340415215547 p T^{2} + \)\(39\!\cdots\!84\)\( p^{2} T^{3} + 15323314340415215547 p^{20} T^{4} + 27035724 p^{39} T^{5} + p^{57} T^{6} \)
13$S_4\times C_2$ \( 1 - 14862401022 T + 47411645745113154327 p T^{2} - \)\(54\!\cdots\!08\)\( p^{2} T^{3} + 47411645745113154327 p^{20} T^{4} - 14862401022 p^{38} T^{5} + p^{57} T^{6} \)
17$S_4\times C_2$ \( 1 - 803332464534 T + \)\(53\!\cdots\!47\)\( p T^{2} - \)\(13\!\cdots\!36\)\( p^{2} T^{3} + \)\(53\!\cdots\!47\)\( p^{20} T^{4} - 803332464534 p^{38} T^{5} + p^{57} T^{6} \)
19$S_4\times C_2$ \( 1 - 3212269666884 T + \)\(86\!\cdots\!81\)\( T^{2} - \)\(13\!\cdots\!68\)\( T^{3} + \)\(86\!\cdots\!81\)\( p^{19} T^{4} - 3212269666884 p^{38} T^{5} + p^{57} T^{6} \)
23$S_4\times C_2$ \( 1 + 24948509305560 T + \)\(41\!\cdots\!93\)\( T^{2} + \)\(41\!\cdots\!72\)\( T^{3} + \)\(41\!\cdots\!93\)\( p^{19} T^{4} + 24948509305560 p^{38} T^{5} + p^{57} T^{6} \)
29$S_4\times C_2$ \( 1 + 77667139511058 T + \)\(11\!\cdots\!63\)\( T^{2} + \)\(90\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!63\)\( p^{19} T^{4} + 77667139511058 p^{38} T^{5} + p^{57} T^{6} \)
31$S_4\times C_2$ \( 1 - 8013926941728 p T + \)\(40\!\cdots\!53\)\( T^{2} - \)\(38\!\cdots\!56\)\( T^{3} + \)\(40\!\cdots\!53\)\( p^{19} T^{4} - 8013926941728 p^{39} T^{5} + p^{57} T^{6} \)
37$S_4\times C_2$ \( 1 - 414866302559142 T + \)\(16\!\cdots\!15\)\( T^{2} - \)\(13\!\cdots\!72\)\( p T^{3} + \)\(16\!\cdots\!15\)\( p^{19} T^{4} - 414866302559142 p^{38} T^{5} + p^{57} T^{6} \)
41$S_4\times C_2$ \( 1 - 2818737880869678 T + \)\(11\!\cdots\!63\)\( T^{2} - \)\(20\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!63\)\( p^{19} T^{4} - 2818737880869678 p^{38} T^{5} + p^{57} T^{6} \)
43$S_4\times C_2$ \( 1 + 6663230715469860 T + \)\(43\!\cdots\!69\)\( T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(43\!\cdots\!69\)\( p^{19} T^{4} + 6663230715469860 p^{38} T^{5} + p^{57} T^{6} \)
47$S_4\times C_2$ \( 1 + 1500497644728624 T + \)\(13\!\cdots\!41\)\( T^{2} + \)\(23\!\cdots\!96\)\( T^{3} + \)\(13\!\cdots\!41\)\( p^{19} T^{4} + 1500497644728624 p^{38} T^{5} + p^{57} T^{6} \)
53$S_4\times C_2$ \( 1 + 56067344774978154 T + \)\(25\!\cdots\!23\)\( T^{2} + \)\(66\!\cdots\!68\)\( T^{3} + \)\(25\!\cdots\!23\)\( p^{19} T^{4} + 56067344774978154 p^{38} T^{5} + p^{57} T^{6} \)
59$S_4\times C_2$ \( 1 + 154317270851496852 T + \)\(16\!\cdots\!97\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!97\)\( p^{19} T^{4} + 154317270851496852 p^{38} T^{5} + p^{57} T^{6} \)
61$S_4\times C_2$ \( 1 - 134994376571654862 T + \)\(15\!\cdots\!43\)\( T^{2} - \)\(13\!\cdots\!84\)\( T^{3} + \)\(15\!\cdots\!43\)\( p^{19} T^{4} - 134994376571654862 p^{38} T^{5} + p^{57} T^{6} \)
67$S_4\times C_2$ \( 1 - 2254211670215676 p T + \)\(12\!\cdots\!49\)\( T^{2} - \)\(11\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!49\)\( p^{19} T^{4} - 2254211670215676 p^{39} T^{5} + p^{57} T^{6} \)
71$S_4\times C_2$ \( 1 + 1210541848845584136 T + \)\(85\!\cdots\!77\)\( T^{2} + \)\(39\!\cdots\!12\)\( T^{3} + \)\(85\!\cdots\!77\)\( p^{19} T^{4} + 1210541848845584136 p^{38} T^{5} + p^{57} T^{6} \)
73$S_4\times C_2$ \( 1 + 81876123599662770 T + \)\(39\!\cdots\!23\)\( T^{2} + \)\(38\!\cdots\!48\)\( T^{3} + \)\(39\!\cdots\!23\)\( p^{19} T^{4} + 81876123599662770 p^{38} T^{5} + p^{57} T^{6} \)
79$S_4\times C_2$ \( 1 + 1439028483035907408 T + \)\(10\!\cdots\!53\)\( T^{2} - \)\(15\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!53\)\( p^{19} T^{4} + 1439028483035907408 p^{38} T^{5} + p^{57} T^{6} \)
83$S_4\times C_2$ \( 1 + 983438102798849916 T + \)\(41\!\cdots\!01\)\( T^{2} + \)\(28\!\cdots\!52\)\( T^{3} + \)\(41\!\cdots\!01\)\( p^{19} T^{4} + 983438102798849916 p^{38} T^{5} + p^{57} T^{6} \)
89$S_4\times C_2$ \( 1 + 1312857528832070946 T + \)\(25\!\cdots\!31\)\( T^{2} + \)\(23\!\cdots\!12\)\( T^{3} + \)\(25\!\cdots\!31\)\( p^{19} T^{4} + 1312857528832070946 p^{38} T^{5} + p^{57} T^{6} \)
97$S_4\times C_2$ \( 1 - 14033245412567998566 T + \)\(22\!\cdots\!51\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(22\!\cdots\!51\)\( p^{19} T^{4} - 14033245412567998566 p^{38} T^{5} + p^{57} T^{6} \)
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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

−10.14786770317220772210210323632, −9.738346294484467617107247036965, −9.723765752978026580548293531918, −9.231456638758503862516499592535, −8.821120039389652744860381741174, −8.185050856038969471843440971178, −8.079831894348866156214380717300, −7.86096827085281386713677807573, −7.53207572438097543149329345648, −6.99991581036445515972319085828, −6.45738131843115598689973997979, −5.98921858281491211161776954935, −5.87303416934868227074482796379, −5.42315734153447438344751182591, −5.02029296498992901362637824968, −4.39218687360617799048962918082, −4.11172311699657198199714333407, −3.45185398061171899615896790091, −3.35594705775134460111887231919, −3.02650596452785634119089805638, −2.62819017148516738164683376870, −2.31690724238159776945604392676, −1.37229596612304277451369327295, −1.33537147769146089989303363900, −1.08337980202909630615424229366, 0, 0, 0, 1.08337980202909630615424229366, 1.33537147769146089989303363900, 1.37229596612304277451369327295, 2.31690724238159776945604392676, 2.62819017148516738164683376870, 3.02650596452785634119089805638, 3.35594705775134460111887231919, 3.45185398061171899615896790091, 4.11172311699657198199714333407, 4.39218687360617799048962918082, 5.02029296498992901362637824968, 5.42315734153447438344751182591, 5.87303416934868227074482796379, 5.98921858281491211161776954935, 6.45738131843115598689973997979, 6.99991581036445515972319085828, 7.53207572438097543149329345648, 7.86096827085281386713677807573, 8.079831894348866156214380717300, 8.185050856038969471843440971178, 8.821120039389652744860381741174, 9.231456638758503862516499592535, 9.723765752978026580548293531918, 9.738346294484467617107247036965, 10.14786770317220772210210323632

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