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Properties

Label 6-7e3-1.1-c11e3-0-0
Degree $6$
Conductor $343$
Sign $1$
Analytic cond. $155.582$
Root an. cond. $2.31913$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 77·2-s − 140·3-s + 2.63e3·4-s + 5.02e3·5-s − 1.07e4·6-s − 5.04e4·7-s + 4.01e4·8-s + 7.33e4·9-s + 3.87e5·10-s − 1.03e6·11-s − 3.69e5·12-s − 1.88e6·13-s − 3.88e6·14-s − 7.03e5·15-s − 5.37e6·16-s + 1.57e7·17-s + 5.64e6·18-s + 1.23e7·19-s + 1.32e7·20-s + 7.05e6·21-s − 8.00e7·22-s + 7.27e6·23-s − 5.62e6·24-s − 2.61e7·25-s − 1.44e8·26-s + 6.74e7·27-s − 1.33e8·28-s + ⋯
L(s)  = 1  + 1.70·2-s − 0.332·3-s + 1.28·4-s + 0.719·5-s − 0.565·6-s − 1.13·7-s + 0.433·8-s + 0.414·9-s + 1.22·10-s − 1.94·11-s − 0.428·12-s − 1.40·13-s − 1.92·14-s − 0.239·15-s − 1.28·16-s + 2.69·17-s + 0.704·18-s + 1.14·19-s + 0.926·20-s + 0.377·21-s − 3.30·22-s + 0.235·23-s − 0.144·24-s − 0.535·25-s − 2.39·26-s + 0.904·27-s − 1.46·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+11/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $1$
Analytic conductor: \(155.582\)
Root analytic conductor: \(2.31913\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 343,\ (\ :11/2, 11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(4.117235028\)
\(L(\frac12)\) \(\approx\) \(4.117235028\)
\(L(\frac{13}{2})\) 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)$
bad7$C_1$ \( ( 1 + p^{5} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 77 T + 1645 p T^{2} - 5643 p^{4} T^{3} + 1645 p^{12} T^{4} - 77 p^{22} T^{5} + p^{33} T^{6} \)
3$S_4\times C_2$ \( 1 + 140 T - 5971 p^{2} T^{2} - 1052632 p^{4} T^{3} - 5971 p^{13} T^{4} + 140 p^{22} T^{5} + p^{33} T^{6} \)
5$S_4\times C_2$ \( 1 - 5026 T + 10282027 p T^{2} - 22355071956 p^{2} T^{3} + 10282027 p^{12} T^{4} - 5026 p^{22} T^{5} + p^{33} T^{6} \)
11$S_4\times C_2$ \( 1 + 1039052 T + 962914998521 T^{2} + 559298310571441992 T^{3} + 962914998521 p^{11} T^{4} + 1039052 p^{22} T^{5} + p^{33} T^{6} \)
13$S_4\times C_2$ \( 1 + 1881222 T + 6164551578399 T^{2} + 6651281953767336284 T^{3} + 6164551578399 p^{11} T^{4} + 1881222 p^{22} T^{5} + p^{33} T^{6} \)
17$S_4\times C_2$ \( 1 - 15797334 T + 168694120042287 T^{2} - \)\(11\!\cdots\!96\)\( T^{3} + 168694120042287 p^{11} T^{4} - 15797334 p^{22} T^{5} + p^{33} T^{6} \)
19$S_4\times C_2$ \( 1 - 12340356 T + 391558323644229 T^{2} - \)\(29\!\cdots\!28\)\( T^{3} + 391558323644229 p^{11} T^{4} - 12340356 p^{22} T^{5} + p^{33} T^{6} \)
23$S_4\times C_2$ \( 1 - 7276752 T - 111678838539051 T^{2} + \)\(28\!\cdots\!64\)\( T^{3} - 111678838539051 p^{11} T^{4} - 7276752 p^{22} T^{5} + p^{33} T^{6} \)
29$S_4\times C_2$ \( 1 + 126853406 T + 30525954562034219 T^{2} + \)\(28\!\cdots\!88\)\( T^{3} + 30525954562034219 p^{11} T^{4} + 126853406 p^{22} T^{5} + p^{33} T^{6} \)
31$S_4\times C_2$ \( 1 - 162184512 T + 60999160202453613 T^{2} - \)\(67\!\cdots\!44\)\( T^{3} + 60999160202453613 p^{11} T^{4} - 162184512 p^{22} T^{5} + p^{33} T^{6} \)
37$S_4\times C_2$ \( 1 - 567121338 T + 570944931389465619 T^{2} - \)\(20\!\cdots\!04\)\( T^{3} + 570944931389465619 p^{11} T^{4} - 567121338 p^{22} T^{5} + p^{33} T^{6} \)
41$S_4\times C_2$ \( 1 - 893682734 T + 972753680000074151 T^{2} - \)\(51\!\cdots\!16\)\( T^{3} + 972753680000074151 p^{11} T^{4} - 893682734 p^{22} T^{5} + p^{33} T^{6} \)
43$S_4\times C_2$ \( 1 - 460197828 T + 2651846066421943833 T^{2} - \)\(83\!\cdots\!28\)\( T^{3} + 2651846066421943833 p^{11} T^{4} - 460197828 p^{22} T^{5} + p^{33} T^{6} \)
47$S_4\times C_2$ \( 1 - 2723825664 T + 8888533789965329661 T^{2} - \)\(13\!\cdots\!32\)\( T^{3} + 8888533789965329661 p^{11} T^{4} - 2723825664 p^{22} T^{5} + p^{33} T^{6} \)
53$S_4\times C_2$ \( 1 - 93426522 T + 22885481722977072339 T^{2} - \)\(53\!\cdots\!56\)\( T^{3} + 22885481722977072339 p^{11} T^{4} - 93426522 p^{22} T^{5} + p^{33} T^{6} \)
59$S_4\times C_2$ \( 1 - 5899174428 T + 95697022136959212333 T^{2} - \)\(35\!\cdots\!44\)\( T^{3} + 95697022136959212333 p^{11} T^{4} - 5899174428 p^{22} T^{5} + p^{33} T^{6} \)
61$S_4\times C_2$ \( 1 - 2106807738 T + 46601731294291872111 T^{2} + \)\(17\!\cdots\!32\)\( T^{3} + 46601731294291872111 p^{11} T^{4} - 2106807738 p^{22} T^{5} + p^{33} T^{6} \)
67$S_4\times C_2$ \( 1 + 27027285348 T + \)\(51\!\cdots\!05\)\( T^{2} + \)\(68\!\cdots\!32\)\( T^{3} + \)\(51\!\cdots\!05\)\( p^{11} T^{4} + 27027285348 p^{22} T^{5} + p^{33} T^{6} \)
71$S_4\times C_2$ \( 1 - 16564049928 T + \)\(69\!\cdots\!49\)\( T^{2} - \)\(70\!\cdots\!84\)\( T^{3} + \)\(69\!\cdots\!49\)\( p^{11} T^{4} - 16564049928 p^{22} T^{5} + p^{33} T^{6} \)
73$S_4\times C_2$ \( 1 - 6036803934 T + \)\(58\!\cdots\!51\)\( T^{2} - \)\(53\!\cdots\!32\)\( T^{3} + \)\(58\!\cdots\!51\)\( p^{11} T^{4} - 6036803934 p^{22} T^{5} + p^{33} T^{6} \)
79$S_4\times C_2$ \( 1 + 54895016736 T + \)\(30\!\cdots\!93\)\( T^{2} + \)\(85\!\cdots\!08\)\( T^{3} + \)\(30\!\cdots\!93\)\( p^{11} T^{4} + 54895016736 p^{22} T^{5} + p^{33} T^{6} \)
83$S_4\times C_2$ \( 1 - 123561203892 T + \)\(86\!\cdots\!17\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!17\)\( p^{11} T^{4} - 123561203892 p^{22} T^{5} + p^{33} T^{6} \)
89$S_4\times C_2$ \( 1 + 91648411106 T + \)\(86\!\cdots\!79\)\( T^{2} + \)\(41\!\cdots\!68\)\( T^{3} + \)\(86\!\cdots\!79\)\( p^{11} T^{4} + 91648411106 p^{22} T^{5} + p^{33} T^{6} \)
97$S_4\times C_2$ \( 1 + 71237114634 T + \)\(18\!\cdots\!83\)\( T^{2} + \)\(99\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!83\)\( p^{11} T^{4} + 71237114634 p^{22} T^{5} + p^{33} T^{6} \)
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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

−17.87263619493243587176081236183, −16.91039736261695984341330220706, −16.47181525738696956065373923618, −15.96736437711924464010517808006, −15.65786370082188780068604824516, −14.77739591748498810225391912774, −14.44658936419028654483534072934, −13.63662388402639273946717675614, −13.49200918853800515981769153242, −12.85521180002906978137767854246, −12.31471395766097859287726181258, −12.12369921771148160859003596871, −11.06478140484470270065222255493, −10.11512455917644728780821437657, −9.944338801048828210142228334072, −9.272375092596959973052351557364, −7.52340847051685921770351915983, −7.52308012507606436635994132236, −6.20169532289873175645940267210, −5.36875841468202894648593385528, −5.30685340563995907518908568756, −4.16125578005830046604543854061, −2.95263408191541634718456541899, −2.54245319684744976085985502355, −0.67236465869310121566944637881, 0.67236465869310121566944637881, 2.54245319684744976085985502355, 2.95263408191541634718456541899, 4.16125578005830046604543854061, 5.30685340563995907518908568756, 5.36875841468202894648593385528, 6.20169532289873175645940267210, 7.52308012507606436635994132236, 7.52340847051685921770351915983, 9.272375092596959973052351557364, 9.944338801048828210142228334072, 10.11512455917644728780821437657, 11.06478140484470270065222255493, 12.12369921771148160859003596871, 12.31471395766097859287726181258, 12.85521180002906978137767854246, 13.49200918853800515981769153242, 13.63662388402639273946717675614, 14.44658936419028654483534072934, 14.77739591748498810225391912774, 15.65786370082188780068604824516, 15.96736437711924464010517808006, 16.47181525738696956065373923618, 16.91039736261695984341330220706, 17.87263619493243587176081236183

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