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Properties

Label 6-2e9-1.1-c19e3-0-0
Degree $6$
Conductor $512$
Sign $1$
Analytic cond. $6133.84$
Root an. cond. $4.27847$
Motivic weight $19$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+19/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $1$
Analytic conductor: \(6133.84\)
Root analytic conductor: \(4.27847\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 512,\ (\ :19/2, 19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(\approx\) \(7.404334550\)
\(L(\frac12)\) \(\approx\) \(7.404334550\)
\(L(\frac{21}{2})\) 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

−14.82524218965554941582576086887, −14.28385632090469457574961782005, −14.12149627631202977851823814370, −13.64192446045102727480616922959, −12.91938590826902732287518888581, −12.36435449026792554925923014208, −11.79951147661603960946027259868, −11.28850260427206759085098384652, −10.69273114928288133077428420694, −10.10034984575813151583727009996, −9.283106713052706848122528979253, −9.108488608272179742106045143723, −8.492543142689373971972239201083, −7.60625812446839756953236334818, −7.46086254826287156262873775633, −6.61136401059012079925460534480, −5.63571925464981809159040619090, −5.13758381464626583717343236826, −4.96225750193500310122462633565, −3.39932494505091093252316442104, −3.23628491705459957840298955472, −2.70422057244818200478374927584, −1.82397421416394743953297479854, −0.946380221864363290003340595567, −0.77624762032002792187370798241, 0.77624762032002792187370798241, 0.946380221864363290003340595567, 1.82397421416394743953297479854, 2.70422057244818200478374927584, 3.23628491705459957840298955472, 3.39932494505091093252316442104, 4.96225750193500310122462633565, 5.13758381464626583717343236826, 5.63571925464981809159040619090, 6.61136401059012079925460534480, 7.46086254826287156262873775633, 7.60625812446839756953236334818, 8.492543142689373971972239201083, 9.108488608272179742106045143723, 9.283106713052706848122528979253, 10.10034984575813151583727009996, 10.69273114928288133077428420694, 11.28850260427206759085098384652, 11.79951147661603960946027259868, 12.36435449026792554925923014208, 12.91938590826902732287518888581, 13.64192446045102727480616922959, 14.12149627631202977851823814370, 14.28385632090469457574961782005, 14.82524218965554941582576086887

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