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Properties

Label 4-3e4-1.1-c27e2-0-1
Degree $4$
Conductor $81$
Sign $1$
Analytic cond. $1727.81$
Root an. cond. $6.44724$
Motivic weight $27$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.15e4·2-s + 1.99e8·4-s + 1.77e9·5-s + 3.69e11·7-s − 1.47e12·8-s − 3.82e13·10-s − 7.57e13·11-s − 1.03e14·13-s − 7.97e15·14-s + 2.32e16·16-s − 3.46e16·17-s + 1.11e17·19-s + 3.54e17·20-s + 1.63e18·22-s + 2.89e18·23-s − 1.25e19·25-s + 2.22e18·26-s + 7.39e19·28-s − 2.99e19·29-s + 1.03e19·31-s − 2.98e20·32-s + 7.48e20·34-s + 6.55e20·35-s − 3.70e21·37-s − 2.40e21·38-s − 2.61e21·40-s − 1.48e21·41-s + ⋯
L(s)  = 1  − 1.86·2-s + 1.48·4-s + 0.649·5-s + 1.44·7-s − 0.949·8-s − 1.20·10-s − 0.661·11-s − 0.0943·13-s − 2.68·14-s + 1.28·16-s − 0.849·17-s + 0.608·19-s + 0.967·20-s + 1.23·22-s + 1.19·23-s − 1.67·25-s + 0.175·26-s + 2.14·28-s − 0.542·29-s + 0.0762·31-s − 1.43·32-s + 1.58·34-s + 0.936·35-s − 2.49·37-s − 1.13·38-s − 0.616·40-s − 0.250·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $1$
Analytic conductor: \(1727.81\)
Root analytic conductor: \(6.44724\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 81,\ (\ :27/2, 27/2),\ 1)\)

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 \( 1 \)
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

−14.54022741485390975140814716239, −13.84278629574104162876813953876, −13.02229668826054183015233517389, −12.00048894620007955653297059122, −11.13926870328619386144314436501, −10.75841916309798125843286245377, −9.681742924252630946263798216162, −9.501372438759515969208985924481, −8.485354307423615132619153791987, −8.134284011057946915261071206896, −7.43365274182317742230896788851, −6.47253430606190516757731408456, −5.35829782773252496915167023141, −4.92295394392876967319009199587, −3.54476327191685594813567837870, −2.48213579864356244850406385888, −1.54239309313523287085748863583, −1.43270305531011933148578041124, 0, 0, 1.43270305531011933148578041124, 1.54239309313523287085748863583, 2.48213579864356244850406385888, 3.54476327191685594813567837870, 4.92295394392876967319009199587, 5.35829782773252496915167023141, 6.47253430606190516757731408456, 7.43365274182317742230896788851, 8.134284011057946915261071206896, 8.485354307423615132619153791987, 9.501372438759515969208985924481, 9.681742924252630946263798216162, 10.75841916309798125843286245377, 11.13926870328619386144314436501, 12.00048894620007955653297059122, 13.02229668826054183015233517389, 13.84278629574104162876813953876, 14.54022741485390975140814716239

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