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Properties

Label 6-3e3-1.1-c25e3-0-0
Degree $6$
Conductor $27$
Sign $1$
Analytic cond. $1676.63$
Root an. cond. $3.44672$
Motivic weight $25$
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  − 3.67e3·2-s + 1.59e6·3-s + 1.80e6·4-s − 1.63e8·5-s − 5.86e9·6-s − 9.62e9·7-s − 3.82e10·8-s + 1.69e12·9-s + 6.00e11·10-s − 5.94e12·11-s + 2.87e12·12-s + 2.48e14·13-s + 3.53e13·14-s − 2.60e14·15-s + 1.07e12·16-s + 6.64e15·17-s − 6.23e15·18-s − 4.28e15·19-s − 2.94e14·20-s − 1.53e16·21-s + 2.18e16·22-s − 5.45e16·23-s − 6.09e16·24-s − 2.91e17·25-s − 9.12e17·26-s + 1.50e18·27-s − 1.73e16·28-s + ⋯
L(s)  = 1  − 0.634·2-s + 1.73·3-s + 0.0538·4-s − 0.298·5-s − 1.09·6-s − 0.262·7-s − 0.196·8-s + 2·9-s + 0.189·10-s − 0.571·11-s + 0.0932·12-s + 2.95·13-s + 0.166·14-s − 0.517·15-s + 0.000953·16-s + 2.76·17-s − 1.26·18-s − 0.443·19-s − 0.0160·20-s − 0.455·21-s + 0.362·22-s − 0.519·23-s − 0.340·24-s − 0.976·25-s − 1.87·26-s + 1.92·27-s − 0.0141·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+25/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $1$
Analytic conductor: \(1676.63\)
Root analytic conductor: \(3.44672\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 27,\ (\ :25/2, 25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(5.087023164\)
\(L(\frac12)\) \(\approx\) \(5.087023164\)
\(L(\frac{27}{2})\) 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)$
bad3$C_1$ \( ( 1 - p^{12} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 1839 p T + 366303 p^{5} T^{2} + 9121989 p^{13} T^{3} + 366303 p^{30} T^{4} + 1839 p^{51} T^{5} + p^{75} T^{6} \)
5$S_4\times C_2$ \( 1 + 1305222 p^{3} T + 508334131259403 p^{4} T^{2} + \)\(80\!\cdots\!56\)\( p^{6} T^{3} + 508334131259403 p^{29} T^{4} + 1305222 p^{53} T^{5} + p^{75} T^{6} \)
7$S_4\times C_2$ \( 1 + 9622572744 T + \)\(18\!\cdots\!95\)\( p T^{2} - \)\(81\!\cdots\!64\)\( p^{4} T^{3} + \)\(18\!\cdots\!95\)\( p^{26} T^{4} + 9622572744 p^{50} T^{5} + p^{75} T^{6} \)
11$S_4\times C_2$ \( 1 + 5946998130780 T + \)\(69\!\cdots\!61\)\( p^{2} T^{2} - \)\(35\!\cdots\!64\)\( p^{4} T^{3} + \)\(69\!\cdots\!61\)\( p^{27} T^{4} + 5946998130780 p^{50} T^{5} + p^{75} T^{6} \)
13$S_4\times C_2$ \( 1 - 248137774407690 T + \)\(36\!\cdots\!11\)\( T^{2} - \)\(26\!\cdots\!84\)\( p T^{3} + \)\(36\!\cdots\!11\)\( p^{25} T^{4} - 248137774407690 p^{50} T^{5} + p^{75} T^{6} \)
17$S_4\times C_2$ \( 1 - 6640885201245174 T + \)\(17\!\cdots\!39\)\( p T^{2} - \)\(29\!\cdots\!32\)\( p^{2} T^{3} + \)\(17\!\cdots\!39\)\( p^{26} T^{4} - 6640885201245174 p^{50} T^{5} + p^{75} T^{6} \)
19$S_4\times C_2$ \( 1 + 4282718959080516 T + \)\(11\!\cdots\!03\)\( p T^{2} + \)\(22\!\cdots\!88\)\( p^{2} T^{3} + \)\(11\!\cdots\!03\)\( p^{26} T^{4} + 4282718959080516 p^{50} T^{5} + p^{75} T^{6} \)
23$S_4\times C_2$ \( 1 + 2372128519633224 p T + \)\(26\!\cdots\!55\)\( p^{3} T^{2} + \)\(99\!\cdots\!56\)\( p^{3} T^{3} + \)\(26\!\cdots\!55\)\( p^{28} T^{4} + 2372128519633224 p^{51} T^{5} + p^{75} T^{6} \)
29$S_4\times C_2$ \( 1 + 506350856671782 T + \)\(37\!\cdots\!67\)\( T^{2} + \)\(69\!\cdots\!36\)\( T^{3} + \)\(37\!\cdots\!67\)\( p^{25} T^{4} + 506350856671782 p^{50} T^{5} + p^{75} T^{6} \)
31$S_4\times C_2$ \( 1 + 120595458951353856 T + \)\(39\!\cdots\!37\)\( T^{2} + \)\(20\!\cdots\!12\)\( T^{3} + \)\(39\!\cdots\!37\)\( p^{25} T^{4} + 120595458951353856 p^{50} T^{5} + p^{75} T^{6} \)
37$S_4\times C_2$ \( 1 - 1922271061255995498 p T + \)\(58\!\cdots\!95\)\( T^{2} - \)\(22\!\cdots\!04\)\( T^{3} + \)\(58\!\cdots\!95\)\( p^{25} T^{4} - 1922271061255995498 p^{51} T^{5} + p^{75} T^{6} \)
41$S_4\times C_2$ \( 1 + \)\(17\!\cdots\!26\)\( T + \)\(32\!\cdots\!87\)\( T^{2} + \)\(53\!\cdots\!72\)\( T^{3} + \)\(32\!\cdots\!87\)\( p^{25} T^{4} + \)\(17\!\cdots\!26\)\( p^{50} T^{5} + p^{75} T^{6} \)
43$S_4\times C_2$ \( 1 - \)\(23\!\cdots\!44\)\( T + \)\(20\!\cdots\!53\)\( T^{2} - \)\(30\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!53\)\( p^{25} T^{4} - \)\(23\!\cdots\!44\)\( p^{50} T^{5} + p^{75} T^{6} \)
47$S_4\times C_2$ \( 1 + \)\(57\!\cdots\!92\)\( T - \)\(89\!\cdots\!63\)\( T^{2} - \)\(50\!\cdots\!40\)\( T^{3} - \)\(89\!\cdots\!63\)\( p^{25} T^{4} + \)\(57\!\cdots\!92\)\( p^{50} T^{5} + p^{75} T^{6} \)
53$S_4\times C_2$ \( 1 - \)\(90\!\cdots\!58\)\( T + \)\(61\!\cdots\!15\)\( T^{2} - \)\(24\!\cdots\!68\)\( T^{3} + \)\(61\!\cdots\!15\)\( p^{25} T^{4} - \)\(90\!\cdots\!58\)\( p^{50} T^{5} + p^{75} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(23\!\cdots\!96\)\( T + \)\(71\!\cdots\!97\)\( T^{2} - \)\(91\!\cdots\!08\)\( T^{3} + \)\(71\!\cdots\!97\)\( p^{25} T^{4} - \)\(23\!\cdots\!96\)\( p^{50} T^{5} + p^{75} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!18\)\( T + \)\(79\!\cdots\!39\)\( T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + \)\(79\!\cdots\!39\)\( p^{25} T^{4} + \)\(11\!\cdots\!18\)\( p^{50} T^{5} + p^{75} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(52\!\cdots\!24\)\( T + \)\(83\!\cdots\!13\)\( T^{2} - \)\(27\!\cdots\!48\)\( T^{3} + \)\(83\!\cdots\!13\)\( p^{25} T^{4} - \)\(52\!\cdots\!24\)\( p^{50} T^{5} + p^{75} T^{6} \)
71$S_4\times C_2$ \( 1 - \)\(26\!\cdots\!36\)\( T + \)\(74\!\cdots\!85\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(74\!\cdots\!85\)\( p^{25} T^{4} - \)\(26\!\cdots\!36\)\( p^{50} T^{5} + p^{75} T^{6} \)
73$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!02\)\( T + \)\(53\!\cdots\!55\)\( T^{2} - \)\(27\!\cdots\!28\)\( T^{3} + \)\(53\!\cdots\!55\)\( p^{25} T^{4} + \)\(11\!\cdots\!02\)\( p^{50} T^{5} + p^{75} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(50\!\cdots\!60\)\( T + \)\(60\!\cdots\!97\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(60\!\cdots\!97\)\( p^{25} T^{4} + \)\(50\!\cdots\!60\)\( p^{50} T^{5} + p^{75} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(59\!\cdots\!84\)\( T + \)\(13\!\cdots\!09\)\( T^{2} + \)\(56\!\cdots\!96\)\( T^{3} + \)\(13\!\cdots\!09\)\( p^{25} T^{4} + \)\(59\!\cdots\!84\)\( p^{50} T^{5} + p^{75} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(33\!\cdots\!66\)\( T + \)\(13\!\cdots\!07\)\( T^{2} + \)\(31\!\cdots\!68\)\( T^{3} + \)\(13\!\cdots\!07\)\( p^{25} T^{4} + \)\(33\!\cdots\!66\)\( p^{50} T^{5} + p^{75} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!66\)\( T + \)\(16\!\cdots\!23\)\( T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(16\!\cdots\!23\)\( p^{25} T^{4} + \)\(12\!\cdots\!66\)\( p^{50} T^{5} + p^{75} T^{6} \)
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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

−18.22844428601110486774267659147, −16.63508741171461474813101967794, −16.41286484262448705433608453651, −15.56311580098218485732615915169, −15.38581705076654848581740693288, −14.29544177172772143879944666284, −14.14382892795298650748192827213, −13.09064735365258600672812454441, −13.07997430333289618967704849630, −11.91066618743116569295894056658, −11.12459044621213012659497332223, −10.12100720464810405860225756743, −9.858869994183767283188236942193, −8.934225425099068132186186830283, −8.326449672783554320021685434952, −8.092931674515688151937542518508, −7.37334657352702505837132554274, −6.21897098066779345805889747051, −5.56555861164984753248591646637, −3.97255800393766658277589207767, −3.62901817443332944087195956465, −3.08526488022681951636320952915, −2.07881602989932182346710711677, −1.23630959236839145675932756156, −0.72979916775864040349047610121, 0.72979916775864040349047610121, 1.23630959236839145675932756156, 2.07881602989932182346710711677, 3.08526488022681951636320952915, 3.62901817443332944087195956465, 3.97255800393766658277589207767, 5.56555861164984753248591646637, 6.21897098066779345805889747051, 7.37334657352702505837132554274, 8.092931674515688151937542518508, 8.326449672783554320021685434952, 8.934225425099068132186186830283, 9.858869994183767283188236942193, 10.12100720464810405860225756743, 11.12459044621213012659497332223, 11.91066618743116569295894056658, 13.07997430333289618967704849630, 13.09064735365258600672812454441, 14.14382892795298650748192827213, 14.29544177172772143879944666284, 15.38581705076654848581740693288, 15.56311580098218485732615915169, 16.41286484262448705433608453651, 16.63508741171461474813101967794, 18.22844428601110486774267659147

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