Properties

Label 4-3e2-1.1-c27e2-0-1
Degree $4$
Conductor $9$
Sign $1$
Analytic cond. $191.979$
Root an. cond. $3.72232$
Motivic weight $27$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.16e3·2-s + 3.18e6·3-s − 1.26e8·4-s − 4.90e9·5-s + 1.01e10·6-s − 1.51e11·7-s − 4.09e11·8-s + 7.62e12·9-s − 1.55e13·10-s + 4.08e13·11-s − 4.04e14·12-s − 4.17e14·13-s − 4.80e14·14-s − 1.56e16·15-s − 6.23e14·16-s − 7.37e16·17-s + 2.41e16·18-s − 3.46e17·19-s + 6.21e17·20-s − 4.83e17·21-s + 1.29e17·22-s + 2.49e18·23-s − 1.30e18·24-s + 8.31e18·25-s − 1.32e18·26-s + 1.62e19·27-s + 1.92e19·28-s + ⋯
L(s)  = 1  + 0.273·2-s + 1.15·3-s − 0.944·4-s − 1.79·5-s + 0.315·6-s − 0.591·7-s − 0.263·8-s + 9-s − 0.491·10-s + 0.356·11-s − 1.09·12-s − 0.382·13-s − 0.161·14-s − 2.07·15-s − 0.0346·16-s − 1.80·17-s + 0.273·18-s − 1.89·19-s + 1.69·20-s − 0.683·21-s + 0.0975·22-s + 1.03·23-s − 0.304·24-s + 1.11·25-s − 0.104·26-s + 0.769·27-s + 0.558·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(191.979\)
Root analytic conductor: \(3.72232\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9,\ (\ :27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{29}{2})\) 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$C_1$ \( ( 1 - p^{13} T )^{2} \)
good2$D_{4}$ \( 1 - 99 p^{5} T + 133597 p^{10} T^{2} - 99 p^{32} T^{3} + p^{54} T^{4} \)
5$D_{4}$ \( 1 + 981213012 p T + 25204374933470302 p^{4} T^{2} + 981213012 p^{28} T^{3} + p^{54} T^{4} \)
7$D_{4}$ \( 1 + 21665298512 p T + \)\(13\!\cdots\!34\)\( p^{2} T^{2} + 21665298512 p^{28} T^{3} + p^{54} T^{4} \)
11$D_{4}$ \( 1 - 3714090193368 p T + \)\(97\!\cdots\!54\)\( p^{2} T^{2} - 3714090193368 p^{28} T^{3} + p^{54} T^{4} \)
13$D_{4}$ \( 1 + 32107530664484 p T + \)\(11\!\cdots\!34\)\( p^{2} T^{2} + 32107530664484 p^{28} T^{3} + p^{54} T^{4} \)
17$D_{4}$ \( 1 + 73795652103164508 T + \)\(27\!\cdots\!86\)\( p T^{2} + 73795652103164508 p^{27} T^{3} + p^{54} T^{4} \)
19$D_{4}$ \( 1 + 346903482355728584 T + \)\(40\!\cdots\!82\)\( p T^{2} + 346903482355728584 p^{27} T^{3} + p^{54} T^{4} \)
23$D_{4}$ \( 1 - 2497625277930684432 T + \)\(48\!\cdots\!58\)\( p T^{2} - 2497625277930684432 p^{27} T^{3} + p^{54} T^{4} \)
29$D_{4}$ \( 1 + 66866817822024885588 T + \)\(64\!\cdots\!38\)\( T^{2} + 66866817822024885588 p^{27} T^{3} + p^{54} T^{4} \)
31$D_{4}$ \( 1 + 4237787486695403168 p T + \)\(32\!\cdots\!22\)\( T^{2} + 4237787486695403168 p^{28} T^{3} + p^{54} T^{4} \)
37$D_{4}$ \( 1 + \)\(24\!\cdots\!04\)\( T + \)\(19\!\cdots\!86\)\( T^{2} + \)\(24\!\cdots\!04\)\( p^{27} T^{3} + p^{54} T^{4} \)
41$D_{4}$ \( 1 - \)\(50\!\cdots\!12\)\( T + \)\(66\!\cdots\!22\)\( T^{2} - \)\(50\!\cdots\!12\)\( p^{27} T^{3} + p^{54} T^{4} \)
43$D_{4}$ \( 1 + \)\(35\!\cdots\!20\)\( T + \)\(57\!\cdots\!90\)\( T^{2} + \)\(35\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
47$D_{4}$ \( 1 - \)\(48\!\cdots\!80\)\( T + \)\(15\!\cdots\!90\)\( T^{2} - \)\(48\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
53$D_{4}$ \( 1 - \)\(35\!\cdots\!92\)\( T + \)\(10\!\cdots\!74\)\( T^{2} - \)\(35\!\cdots\!92\)\( p^{27} T^{3} + p^{54} T^{4} \)
59$D_{4}$ \( 1 + \)\(15\!\cdots\!76\)\( T + \)\(17\!\cdots\!58\)\( T^{2} + \)\(15\!\cdots\!76\)\( p^{27} T^{3} + p^{54} T^{4} \)
61$D_{4}$ \( 1 + \)\(29\!\cdots\!00\)\( T + \)\(30\!\cdots\!98\)\( T^{2} + \)\(29\!\cdots\!00\)\( p^{27} T^{3} + p^{54} T^{4} \)
67$D_{4}$ \( 1 + \)\(14\!\cdots\!88\)\( T + \)\(40\!\cdots\!82\)\( T^{2} + \)\(14\!\cdots\!88\)\( p^{27} T^{3} + p^{54} T^{4} \)
71$D_{4}$ \( 1 + \)\(18\!\cdots\!76\)\( T + \)\(23\!\cdots\!26\)\( T^{2} + \)\(18\!\cdots\!76\)\( p^{27} T^{3} + p^{54} T^{4} \)
73$D_{4}$ \( 1 + \)\(18\!\cdots\!88\)\( T + \)\(33\!\cdots\!54\)\( T^{2} + \)\(18\!\cdots\!88\)\( p^{27} T^{3} + p^{54} T^{4} \)
79$D_{4}$ \( 1 + \)\(29\!\cdots\!40\)\( T + \)\(21\!\cdots\!18\)\( T^{2} + \)\(29\!\cdots\!40\)\( p^{27} T^{3} + p^{54} T^{4} \)
83$D_{4}$ \( 1 - \)\(68\!\cdots\!36\)\( T + \)\(13\!\cdots\!22\)\( T^{2} - \)\(68\!\cdots\!36\)\( p^{27} T^{3} + p^{54} T^{4} \)
89$D_{4}$ \( 1 - \)\(52\!\cdots\!36\)\( T + \)\(14\!\cdots\!98\)\( T^{2} - \)\(52\!\cdots\!36\)\( p^{27} T^{3} + p^{54} T^{4} \)
97$D_{4}$ \( 1 - \)\(79\!\cdots\!12\)\( T + \)\(68\!\cdots\!62\)\( T^{2} - \)\(79\!\cdots\!12\)\( p^{27} T^{3} + p^{54} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.08900402245685480414177577479, −18.48578427542926461750281328292, −17.12264849363198900809883350514, −16.00015634475070387586438232995, −14.99663130486949289666488887069, −14.87340252181252745824408020535, −13.21909217079641720774820870763, −13.11552639560625277361316242503, −11.78413174033192768519422605877, −10.64752720863661838682538894663, −8.929021207689181822044436972339, −8.905185611610231180096836905380, −7.63813961817774875535686408725, −6.72817887969084872788180468735, −4.58973508480773331284927121023, −4.10253007810715135890686406909, −3.35080718784511089109607837182, −2.03829725232267675684035513441, 0, 0, 2.03829725232267675684035513441, 3.35080718784511089109607837182, 4.10253007810715135890686406909, 4.58973508480773331284927121023, 6.72817887969084872788180468735, 7.63813961817774875535686408725, 8.905185611610231180096836905380, 8.929021207689181822044436972339, 10.64752720863661838682538894663, 11.78413174033192768519422605877, 13.11552639560625277361316242503, 13.21909217079641720774820870763, 14.87340252181252745824408020535, 14.99663130486949289666488887069, 16.00015634475070387586438232995, 17.12264849363198900809883350514, 18.48578427542926461750281328292, 19.08900402245685480414177577479

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