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Properties

Label 4-3e2-1.1-c27e2-0-0
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.15e4·2-s − 3.18e6·3-s + 1.99e8·4-s − 1.77e9·5-s − 6.88e10·6-s + 3.69e11·7-s + 1.47e12·8-s + 7.62e12·9-s − 3.82e13·10-s + 7.57e13·11-s − 6.37e14·12-s − 1.03e14·13-s + 7.97e15·14-s + 5.65e15·15-s + 2.32e16·16-s + 3.46e16·17-s + 1.64e17·18-s + 1.11e17·19-s − 3.54e17·20-s − 1.17e18·21-s + 1.63e18·22-s − 2.89e18·23-s − 4.70e18·24-s − 1.25e19·25-s − 2.22e18·26-s − 1.62e19·27-s + 7.39e19·28-s + ⋯
L(s)  = 1  + 1.86·2-s − 1.15·3-s + 1.48·4-s − 0.649·5-s − 2.15·6-s + 1.44·7-s + 0.949·8-s + 9-s − 1.20·10-s + 0.661·11-s − 1.72·12-s − 0.0943·13-s + 2.68·14-s + 0.749·15-s + 1.28·16-s + 0.849·17-s + 1.86·18-s + 0.608·19-s − 0.967·20-s − 1.66·21-s + 1.23·22-s − 1.19·23-s − 1.09·24-s − 1.67·25-s − 0.175·26-s − 0.769·27-s + 2.14·28-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: \(0\)
Selberg data: \((4,\ 9,\ (\ :27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(5.048440310\)
\(L(\frac12)\) \(\approx\) \(5.048440310\)
\(L(\frac{29}{2})\) 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$D_{4}$ \( 1 - 10791 p T + 4153237 p^{6} T^{2} - 10791 p^{28} T^{3} + p^{54} T^{4} \)
5$D_{4}$ \( 1 + 70877844 p^{2} T + 5007752027691686 p^{5} T^{2} + 70877844 p^{29} T^{3} + p^{54} T^{4} \)
7$D_{4}$ \( 1 - 52809314272 p T + \)\(25\!\cdots\!94\)\( p^{2} T^{2} - 52809314272 p^{28} T^{3} + p^{54} T^{4} \)
11$D_{4}$ \( 1 - 75762335668248 T + \)\(16\!\cdots\!14\)\( p^{2} T^{2} - 75762335668248 p^{27} T^{3} + p^{54} T^{4} \)
13$D_{4}$ \( 1 + 7924698398276 p T + \)\(46\!\cdots\!54\)\( p^{2} T^{2} + 7924698398276 p^{28} T^{3} + p^{54} T^{4} \)
17$D_{4}$ \( 1 - 2040683034067524 p T + \)\(29\!\cdots\!58\)\( p^{2} T^{2} - 2040683034067524 p^{28} T^{3} + p^{54} T^{4} \)
19$D_{4}$ \( 1 - 5874761924139064 p T + \)\(36\!\cdots\!82\)\( p T^{2} - 5874761924139064 p^{28} T^{3} + p^{54} T^{4} \)
23$D_{4}$ \( 1 + 125771438079103824 p T + \)\(18\!\cdots\!26\)\( p^{2} T^{2} + 125771438079103824 p^{28} T^{3} + p^{54} T^{4} \)
29$D_{4}$ \( 1 - 29959552473322806972 T + \)\(45\!\cdots\!78\)\( T^{2} - 29959552473322806972 p^{27} T^{3} + p^{54} T^{4} \)
31$D_{4}$ \( 1 - 10367463257055494032 T + \)\(26\!\cdots\!22\)\( T^{2} - 10367463257055494032 p^{27} T^{3} + p^{54} T^{4} \)
37$D_{4}$ \( 1 + \)\(37\!\cdots\!16\)\( T + \)\(78\!\cdots\!86\)\( T^{2} + \)\(37\!\cdots\!16\)\( p^{27} T^{3} + p^{54} T^{4} \)
41$D_{4}$ \( 1 - \)\(14\!\cdots\!72\)\( T + \)\(35\!\cdots\!42\)\( T^{2} - \)\(14\!\cdots\!72\)\( p^{27} T^{3} + p^{54} T^{4} \)
43$D_{4}$ \( 1 - \)\(97\!\cdots\!40\)\( T + \)\(26\!\cdots\!90\)\( T^{2} - \)\(97\!\cdots\!40\)\( p^{27} T^{3} + p^{54} T^{4} \)
47$D_{4}$ \( 1 - \)\(89\!\cdots\!20\)\( T + \)\(44\!\cdots\!90\)\( T^{2} - \)\(89\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
53$D_{4}$ \( 1 - \)\(42\!\cdots\!48\)\( T + \)\(11\!\cdots\!94\)\( T^{2} - \)\(42\!\cdots\!48\)\( p^{27} T^{3} + p^{54} T^{4} \)
59$D_{4}$ \( 1 - \)\(36\!\cdots\!24\)\( T + \)\(55\!\cdots\!38\)\( T^{2} - \)\(36\!\cdots\!24\)\( p^{27} T^{3} + p^{54} T^{4} \)
61$D_{4}$ \( 1 - \)\(92\!\cdots\!20\)\( T + \)\(32\!\cdots\!58\)\( T^{2} - \)\(92\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
67$D_{4}$ \( 1 - \)\(41\!\cdots\!68\)\( T + \)\(44\!\cdots\!02\)\( T^{2} - \)\(41\!\cdots\!68\)\( p^{27} T^{3} + p^{54} T^{4} \)
71$D_{4}$ \( 1 - \)\(97\!\cdots\!84\)\( T + \)\(15\!\cdots\!46\)\( T^{2} - \)\(97\!\cdots\!84\)\( p^{27} T^{3} + p^{54} T^{4} \)
73$D_{4}$ \( 1 - \)\(28\!\cdots\!88\)\( T + \)\(56\!\cdots\!14\)\( T^{2} - \)\(28\!\cdots\!88\)\( p^{27} T^{3} + p^{54} T^{4} \)
79$D_{4}$ \( 1 + \)\(49\!\cdots\!20\)\( T + \)\(34\!\cdots\!18\)\( T^{2} + \)\(49\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
83$D_{4}$ \( 1 + \)\(12\!\cdots\!36\)\( T + \)\(16\!\cdots\!82\)\( T^{2} + \)\(12\!\cdots\!36\)\( p^{27} T^{3} + p^{54} T^{4} \)
89$D_{4}$ \( 1 - \)\(20\!\cdots\!56\)\( T + \)\(40\!\cdots\!38\)\( T^{2} - \)\(20\!\cdots\!56\)\( p^{27} T^{3} + p^{54} T^{4} \)
97$D_{4}$ \( 1 + \)\(14\!\cdots\!72\)\( T + \)\(13\!\cdots\!22\)\( T^{2} + \)\(14\!\cdots\!72\)\( p^{27} T^{3} + p^{54} T^{4} \)
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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

−20.03702430900785282319792107584, −18.99111174389490532947712456263, −17.75290237180002296568425167393, −17.16706581868042150940126326892, −16.01085138427895718668024794883, −15.24382046246837735142537072943, −14.08355488248378304772540539740, −13.88711071268146075792173720516, −12.17052150369082891496338677350, −12.13461840817885297569941909304, −11.22625307050688578883812168274, −10.13118835683720197053931014935, −8.105919284656919411437338750971, −7.15881120904364726585389400392, −5.63613204169456285987627106693, −5.30940080005576982773030934164, −4.15308581087043677264637868676, −3.85754340835505704010694964980, −1.86461830146449799650916188094, −0.799486610718465007490355893782, 0.799486610718465007490355893782, 1.86461830146449799650916188094, 3.85754340835505704010694964980, 4.15308581087043677264637868676, 5.30940080005576982773030934164, 5.63613204169456285987627106693, 7.15881120904364726585389400392, 8.105919284656919411437338750971, 10.13118835683720197053931014935, 11.22625307050688578883812168274, 12.13461840817885297569941909304, 12.17052150369082891496338677350, 13.88711071268146075792173720516, 14.08355488248378304772540539740, 15.24382046246837735142537072943, 16.01085138427895718668024794883, 17.16706581868042150940126326892, 17.75290237180002296568425167393, 18.99111174389490532947712456263, 20.03702430900785282319792107584

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