Properties

Degree 8
Conductor $ 3^{4} $
Sign $1$
Motivic weight 37
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.37e5·2-s + 1.54e9·3-s − 6.09e9·4-s − 4.09e12·5-s + 6.78e14·6-s + 6.60e15·7-s − 3.81e15·8-s + 1.50e18·9-s − 1.79e18·10-s + 2.09e19·11-s − 9.45e18·12-s + 5.18e19·13-s + 2.89e21·14-s − 6.35e21·15-s + 1.05e22·16-s + 8.11e22·17-s + 6.56e23·18-s − 5.44e23·19-s + 2.50e22·20-s + 1.02e25·21-s + 9.16e24·22-s − 6.10e24·23-s − 5.90e24·24-s − 8.94e25·25-s + 2.26e25·26-s + 1.16e27·27-s − 4.02e25·28-s + ⋯
L(s)  = 1  + 1.18·2-s + 2.30·3-s − 0.0443·4-s − 0.480·5-s + 2.72·6-s + 1.53·7-s − 0.0748·8-s + 10/3·9-s − 0.567·10-s + 1.13·11-s − 0.102·12-s + 0.127·13-s + 1.80·14-s − 1.10·15-s + 0.560·16-s + 1.39·17-s + 3.93·18-s − 1.19·19-s + 0.0213·20-s + 3.54·21-s + 1.34·22-s − 0.392·23-s − 0.172·24-s − 1.22·25-s + 0.150·26-s + 3.84·27-s − 0.0680·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+37/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(37\)
character  :  induced by $\chi_{3} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 81,\ (\ :37/2, 37/2, 37/2, 37/2),\ 1)$
$L(19)$  $\approx$  $46.45433381$
$L(\frac12)$  $\approx$  $46.45433381$
$L(\frac{39}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p(T)\) is a polynomial of degree 8. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - p^{18} T )^{4} \)
good2$C_2 \wr S_4$ \( 1 - 218781 p T + 6173708785 p^{5} T^{2} - 2603177122851 p^{15} T^{3} + 783039221777613 p^{25} T^{4} - 2603177122851 p^{52} T^{5} + 6173708785 p^{79} T^{6} - 218781 p^{112} T^{7} + p^{148} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 4099829756904 T + \)\(84\!\cdots\!16\)\( p^{3} T^{2} - \)\(16\!\cdots\!48\)\( p^{8} T^{3} + \)\(34\!\cdots\!82\)\( p^{14} T^{4} - \)\(16\!\cdots\!48\)\( p^{45} T^{5} + \)\(84\!\cdots\!16\)\( p^{77} T^{6} + 4099829756904 p^{111} T^{7} + p^{148} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 6605809948153184 T + \)\(55\!\cdots\!76\)\( p T^{2} - \)\(69\!\cdots\!44\)\( p^{4} T^{3} + \)\(65\!\cdots\!50\)\( p^{7} T^{4} - \)\(69\!\cdots\!44\)\( p^{41} T^{5} + \)\(55\!\cdots\!76\)\( p^{75} T^{6} - 6605809948153184 p^{111} T^{7} + p^{148} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 1904882622926844816 p T + \)\(76\!\cdots\!52\)\( p^{2} T^{2} - \)\(80\!\cdots\!20\)\( p^{5} T^{3} + \)\(19\!\cdots\!26\)\( p^{8} T^{4} - \)\(80\!\cdots\!20\)\( p^{42} T^{5} + \)\(76\!\cdots\!52\)\( p^{76} T^{6} - 1904882622926844816 p^{112} T^{7} + p^{148} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 51830892788989874168 T + \)\(47\!\cdots\!04\)\( T^{2} - \)\(13\!\cdots\!52\)\( p T^{3} + \)\(48\!\cdots\!50\)\( p^{3} T^{4} - \)\(13\!\cdots\!52\)\( p^{38} T^{5} + \)\(47\!\cdots\!04\)\( p^{74} T^{6} - 51830892788989874168 p^{111} T^{7} + p^{148} T^{8} \)
17$C_2 \wr S_4$ \( 1 - \)\(81\!\cdots\!28\)\( T + \)\(59\!\cdots\!56\)\( p T^{2} - \)\(14\!\cdots\!80\)\( p^{3} T^{3} + \)\(31\!\cdots\!18\)\( p^{5} T^{4} - \)\(14\!\cdots\!80\)\( p^{40} T^{5} + \)\(59\!\cdots\!56\)\( p^{75} T^{6} - \)\(81\!\cdots\!28\)\( p^{111} T^{7} + p^{148} T^{8} \)
19$C_2 \wr S_4$ \( 1 + \)\(54\!\cdots\!32\)\( T + \)\(55\!\cdots\!72\)\( T^{2} + \)\(13\!\cdots\!16\)\( p T^{3} + \)\(44\!\cdots\!34\)\( p^{2} T^{4} + \)\(13\!\cdots\!16\)\( p^{38} T^{5} + \)\(55\!\cdots\!72\)\( p^{74} T^{6} + \)\(54\!\cdots\!32\)\( p^{111} T^{7} + p^{148} T^{8} \)
23$C_2 \wr S_4$ \( 1 + \)\(61\!\cdots\!88\)\( T + \)\(42\!\cdots\!16\)\( p T^{2} + \)\(32\!\cdots\!16\)\( p^{3} T^{3} + \)\(16\!\cdots\!70\)\( p^{3} T^{4} + \)\(32\!\cdots\!16\)\( p^{40} T^{5} + \)\(42\!\cdots\!16\)\( p^{75} T^{6} + \)\(61\!\cdots\!88\)\( p^{111} T^{7} + p^{148} T^{8} \)
29$C_2 \wr S_4$ \( 1 - \)\(41\!\cdots\!36\)\( T + \)\(35\!\cdots\!20\)\( p T^{2} - \)\(20\!\cdots\!52\)\( p^{2} T^{3} + \)\(89\!\cdots\!02\)\( p^{3} T^{4} - \)\(20\!\cdots\!52\)\( p^{39} T^{5} + \)\(35\!\cdots\!20\)\( p^{75} T^{6} - \)\(41\!\cdots\!36\)\( p^{111} T^{7} + p^{148} T^{8} \)
31$C_2 \wr S_4$ \( 1 - \)\(89\!\cdots\!64\)\( T + \)\(23\!\cdots\!28\)\( p T^{2} - \)\(38\!\cdots\!92\)\( p^{2} T^{3} + \)\(57\!\cdots\!94\)\( p^{3} T^{4} - \)\(38\!\cdots\!92\)\( p^{39} T^{5} + \)\(23\!\cdots\!28\)\( p^{75} T^{6} - \)\(89\!\cdots\!64\)\( p^{111} T^{7} + p^{148} T^{8} \)
37$C_2 \wr S_4$ \( 1 - \)\(55\!\cdots\!24\)\( T + \)\(22\!\cdots\!72\)\( T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} - \)\(20\!\cdots\!44\)\( p^{37} T^{5} + \)\(22\!\cdots\!72\)\( p^{74} T^{6} - \)\(55\!\cdots\!24\)\( p^{111} T^{7} + p^{148} T^{8} \)
41$C_2 \wr S_4$ \( 1 + \)\(86\!\cdots\!76\)\( T + \)\(17\!\cdots\!88\)\( T^{2} + \)\(10\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!34\)\( T^{4} + \)\(10\!\cdots\!28\)\( p^{37} T^{5} + \)\(17\!\cdots\!88\)\( p^{74} T^{6} + \)\(86\!\cdots\!76\)\( p^{111} T^{7} + p^{148} T^{8} \)
43$C_2 \wr S_4$ \( 1 + \)\(50\!\cdots\!80\)\( T + \)\(85\!\cdots\!80\)\( T^{2} + \)\(56\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!98\)\( T^{4} + \)\(56\!\cdots\!40\)\( p^{37} T^{5} + \)\(85\!\cdots\!80\)\( p^{74} T^{6} + \)\(50\!\cdots\!80\)\( p^{111} T^{7} + p^{148} T^{8} \)
47$C_2 \wr S_4$ \( 1 - \)\(42\!\cdots\!20\)\( T + \)\(24\!\cdots\!20\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!40\)\( p^{37} T^{5} + \)\(24\!\cdots\!20\)\( p^{74} T^{6} - \)\(42\!\cdots\!20\)\( p^{111} T^{7} + p^{148} T^{8} \)
53$C_2 \wr S_4$ \( 1 + \)\(12\!\cdots\!88\)\( T + \)\(20\!\cdots\!88\)\( T^{2} + \)\(19\!\cdots\!32\)\( T^{3} + \)\(18\!\cdots\!50\)\( T^{4} + \)\(19\!\cdots\!32\)\( p^{37} T^{5} + \)\(20\!\cdots\!88\)\( p^{74} T^{6} + \)\(12\!\cdots\!88\)\( p^{111} T^{7} + p^{148} T^{8} \)
59$C_2 \wr S_4$ \( 1 + \)\(13\!\cdots\!88\)\( T + \)\(62\!\cdots\!12\)\( T^{2} - \)\(19\!\cdots\!84\)\( T^{3} - \)\(29\!\cdots\!66\)\( T^{4} - \)\(19\!\cdots\!84\)\( p^{37} T^{5} + \)\(62\!\cdots\!12\)\( p^{74} T^{6} + \)\(13\!\cdots\!88\)\( p^{111} T^{7} + p^{148} T^{8} \)
61$C_2 \wr S_4$ \( 1 + \)\(12\!\cdots\!60\)\( T + \)\(43\!\cdots\!16\)\( T^{2} + \)\(36\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!46\)\( T^{4} + \)\(36\!\cdots\!40\)\( p^{37} T^{5} + \)\(43\!\cdots\!16\)\( p^{74} T^{6} + \)\(12\!\cdots\!60\)\( p^{111} T^{7} + p^{148} T^{8} \)
67$C_2 \wr S_4$ \( 1 - \)\(16\!\cdots\!48\)\( T + \)\(20\!\cdots\!72\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!66\)\( T^{4} - \)\(16\!\cdots\!00\)\( p^{37} T^{5} + \)\(20\!\cdots\!72\)\( p^{74} T^{6} - \)\(16\!\cdots\!48\)\( p^{111} T^{7} + p^{148} T^{8} \)
71$C_2 \wr S_4$ \( 1 - \)\(10\!\cdots\!88\)\( T + \)\(55\!\cdots\!68\)\( T^{2} - \)\(47\!\cdots\!16\)\( T^{3} + \)\(14\!\cdots\!70\)\( T^{4} - \)\(47\!\cdots\!16\)\( p^{37} T^{5} + \)\(55\!\cdots\!68\)\( p^{74} T^{6} - \)\(10\!\cdots\!88\)\( p^{111} T^{7} + p^{148} T^{8} \)
73$C_2 \wr S_4$ \( 1 + \)\(19\!\cdots\!48\)\( T + \)\(28\!\cdots\!68\)\( T^{2} + \)\(45\!\cdots\!52\)\( T^{3} + \)\(35\!\cdots\!90\)\( T^{4} + \)\(45\!\cdots\!52\)\( p^{37} T^{5} + \)\(28\!\cdots\!68\)\( p^{74} T^{6} + \)\(19\!\cdots\!48\)\( p^{111} T^{7} + p^{148} T^{8} \)
79$C_2 \wr S_4$ \( 1 - \)\(42\!\cdots\!20\)\( T + \)\(83\!\cdots\!36\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} - \)\(27\!\cdots\!14\)\( T^{4} - \)\(13\!\cdots\!40\)\( p^{37} T^{5} + \)\(83\!\cdots\!36\)\( p^{74} T^{6} - \)\(42\!\cdots\!20\)\( p^{111} T^{7} + p^{148} T^{8} \)
83$C_2 \wr S_4$ \( 1 + \)\(46\!\cdots\!24\)\( T + \)\(47\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!44\)\( T^{3} + \)\(74\!\cdots\!46\)\( T^{4} + \)\(14\!\cdots\!44\)\( p^{37} T^{5} + \)\(47\!\cdots\!60\)\( p^{74} T^{6} + \)\(46\!\cdots\!24\)\( p^{111} T^{7} + p^{148} T^{8} \)
89$C_2 \wr S_4$ \( 1 + \)\(31\!\cdots\!52\)\( T + \)\(35\!\cdots\!52\)\( T^{2} + \)\(89\!\cdots\!84\)\( T^{3} + \)\(60\!\cdots\!34\)\( T^{4} + \)\(89\!\cdots\!84\)\( p^{37} T^{5} + \)\(35\!\cdots\!52\)\( p^{74} T^{6} + \)\(31\!\cdots\!52\)\( p^{111} T^{7} + p^{148} T^{8} \)
97$C_2 \wr S_4$ \( 1 - \)\(44\!\cdots\!48\)\( T + \)\(13\!\cdots\!12\)\( T^{2} - \)\(24\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!40\)\( p^{37} T^{5} + \)\(13\!\cdots\!12\)\( p^{74} T^{6} - \)\(44\!\cdots\!48\)\( p^{111} T^{7} + p^{148} T^{8} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.48591441628306933787316764005, −11.18580836267465425557109965037, −10.11908614912665704922563944539, −10.03059963989914318847138277500, −9.831991251053794891370908828608, −8.712982752843244742574982245931, −8.692730932069098079164912724810, −8.201514534597191258071216342069, −7.917206073367492893260757672301, −7.76577579717934416815426708984, −6.94441566309081609996071473436, −6.29938234778421846415242911055, −6.13792801550831622399659488265, −4.87642316070806856068622124168, −4.74024898788082779304229807135, −4.42275179296483521805118028822, −4.25974196820357533378371411840, −3.60206902088349860274362676812, −3.14887007459806955539309380437, −2.97816892620446375421952644036, −2.24733023595259763119491346067, −1.77757456369236737958588902134, −1.48277362283597403311602540510, −0.936580817219163961350976004255, −0.65964473813392773060173476615, 0.65964473813392773060173476615, 0.936580817219163961350976004255, 1.48277362283597403311602540510, 1.77757456369236737958588902134, 2.24733023595259763119491346067, 2.97816892620446375421952644036, 3.14887007459806955539309380437, 3.60206902088349860274362676812, 4.25974196820357533378371411840, 4.42275179296483521805118028822, 4.74024898788082779304229807135, 4.87642316070806856068622124168, 6.13792801550831622399659488265, 6.29938234778421846415242911055, 6.94441566309081609996071473436, 7.76577579717934416815426708984, 7.917206073367492893260757672301, 8.201514534597191258071216342069, 8.692730932069098079164912724810, 8.712982752843244742574982245931, 9.831991251053794891370908828608, 10.03059963989914318847138277500, 10.11908614912665704922563944539, 11.18580836267465425557109965037, 11.48591441628306933787316764005

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