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Properties

Label 4-18e2-1.1-c27e2-0-2
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $6911.26$
Root an. cond. $9.11778$
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  + 1.63e4·2-s + 2.01e8·4-s + 1.07e9·5-s + 9.83e10·7-s + 2.19e12·8-s + 1.76e13·10-s − 1.11e14·11-s − 1.51e15·13-s + 1.61e15·14-s + 2.25e16·16-s − 3.80e16·17-s − 5.66e16·19-s + 2.16e17·20-s − 1.83e18·22-s − 2.94e18·23-s − 6.04e18·25-s − 2.48e19·26-s + 1.98e19·28-s − 2.63e19·29-s + 1.56e19·31-s + 2.21e20·32-s − 6.23e20·34-s + 1.05e20·35-s − 1.55e21·37-s − 9.28e20·38-s + 2.36e21·40-s + 2.80e21·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.393·5-s + 0.383·7-s + 1.41·8-s + 0.556·10-s − 0.976·11-s − 1.38·13-s + 0.542·14-s + 5/4·16-s − 0.931·17-s − 0.309·19-s + 0.590·20-s − 1.38·22-s − 1.21·23-s − 0.811·25-s − 1.96·26-s + 0.575·28-s − 0.476·29-s + 0.115·31-s + 1.06·32-s − 1.31·34-s + 0.151·35-s − 1.04·37-s − 0.437·38-s + 0.556·40-s + 0.473·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(6911.26\)
Root analytic conductor: \(9.11778\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 324,\ (\ :27/2, 27/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 - p^{13} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 - 42988704 p^{2} T + 2305493080823186 p^{5} T^{2} - 42988704 p^{29} T^{3} + p^{54} T^{4} \)
7$D_{4}$ \( 1 - 14053514200 p T + \)\(24\!\cdots\!10\)\( p^{2} T^{2} - 14053514200 p^{28} T^{3} + p^{54} T^{4} \)
11$D_{4}$ \( 1 + 923927233920 p^{2} T + \)\(11\!\cdots\!26\)\( p^{2} T^{2} + 923927233920 p^{29} T^{3} + p^{54} T^{4} \)
13$D_{4}$ \( 1 + 116739965357180 p T + \)\(11\!\cdots\!10\)\( p^{2} T^{2} + 116739965357180 p^{28} T^{3} + p^{54} T^{4} \)
17$D_{4}$ \( 1 + 38054810425611456 T + \)\(17\!\cdots\!90\)\( p T^{2} + 38054810425611456 p^{27} T^{3} + p^{54} T^{4} \)
19$D_{4}$ \( 1 + 56646249550061072 T + \)\(41\!\cdots\!46\)\( p T^{2} + 56646249550061072 p^{27} T^{3} + p^{54} T^{4} \)
23$D_{4}$ \( 1 + 2942542507293523200 T + \)\(59\!\cdots\!78\)\( p T^{2} + 2942542507293523200 p^{27} T^{3} + p^{54} T^{4} \)
29$D_{4}$ \( 1 + 26313378910948644000 T + \)\(58\!\cdots\!62\)\( T^{2} + 26313378910948644000 p^{27} T^{3} + p^{54} T^{4} \)
31$D_{4}$ \( 1 - 15634841744496978472 T + \)\(23\!\cdots\!18\)\( T^{2} - 15634841744496978472 p^{27} T^{3} + p^{54} T^{4} \)
37$D_{4}$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(28\!\cdots\!90\)\( T^{2} + \)\(15\!\cdots\!00\)\( p^{27} T^{3} + p^{54} T^{4} \)
41$D_{4}$ \( 1 - \)\(28\!\cdots\!40\)\( T + \)\(32\!\cdots\!62\)\( T^{2} - \)\(28\!\cdots\!40\)\( p^{27} T^{3} + p^{54} T^{4} \)
43$D_{4}$ \( 1 + \)\(20\!\cdots\!20\)\( T + \)\(33\!\cdots\!78\)\( T^{2} + \)\(20\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
47$D_{4}$ \( 1 + \)\(38\!\cdots\!00\)\( T + \)\(31\!\cdots\!26\)\( T^{2} + \)\(38\!\cdots\!00\)\( p^{27} T^{3} + p^{54} T^{4} \)
53$D_{4}$ \( 1 - \)\(79\!\cdots\!76\)\( T + \)\(67\!\cdots\!18\)\( T^{2} - \)\(79\!\cdots\!76\)\( p^{27} T^{3} + p^{54} T^{4} \)
59$D_{4}$ \( 1 - \)\(28\!\cdots\!60\)\( T + \)\(13\!\cdots\!62\)\( T^{2} - \)\(28\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
61$D_{4}$ \( 1 - \)\(10\!\cdots\!24\)\( T + \)\(15\!\cdots\!86\)\( T^{2} - \)\(10\!\cdots\!24\)\( p^{27} T^{3} + p^{54} T^{4} \)
67$D_{4}$ \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(23\!\cdots\!42\)\( T^{2} - \)\(10\!\cdots\!00\)\( p^{27} T^{3} + p^{54} T^{4} \)
71$D_{4}$ \( 1 - \)\(27\!\cdots\!60\)\( T - \)\(15\!\cdots\!42\)\( T^{2} - \)\(27\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
73$D_{4}$ \( 1 - \)\(27\!\cdots\!00\)\( T + \)\(59\!\cdots\!94\)\( T^{2} - \)\(27\!\cdots\!00\)\( p^{27} T^{3} + p^{54} T^{4} \)
79$D_{4}$ \( 1 - \)\(42\!\cdots\!88\)\( T + \)\(26\!\cdots\!54\)\( T^{2} - \)\(42\!\cdots\!88\)\( p^{27} T^{3} + p^{54} T^{4} \)
83$D_{4}$ \( 1 + \)\(17\!\cdots\!12\)\( T + \)\(20\!\cdots\!90\)\( T^{2} + \)\(17\!\cdots\!12\)\( p^{27} T^{3} + p^{54} T^{4} \)
89$D_{4}$ \( 1 + \)\(64\!\cdots\!80\)\( T + \)\(18\!\cdots\!74\)\( T^{2} + \)\(64\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
97$D_{4}$ \( 1 + \)\(35\!\cdots\!20\)\( T + \)\(59\!\cdots\!10\)\( T^{2} + \)\(35\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
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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

−12.50043296467404237939544727902, −12.11294677755526756731381454911, −11.32089418591034566504780451062, −10.86343315491896409451241649589, −9.898035679329563907669108234156, −9.713899133429240960166209407847, −8.175715778102382297594997304705, −8.063775840951172586972464087393, −6.88272937685000054546244148602, −6.70592469241105562929820761127, −5.51349304415197470893758061348, −5.38552641139121103578920913728, −4.59544687387231346152163556826, −4.07741276192050139284940616748, −3.21930049843278884392856896604, −2.48653765387048409152262317229, −2.03640440029413771850585145967, −1.54602073030053070428758453437, 0, 0, 1.54602073030053070428758453437, 2.03640440029413771850585145967, 2.48653765387048409152262317229, 3.21930049843278884392856896604, 4.07741276192050139284940616748, 4.59544687387231346152163556826, 5.38552641139121103578920913728, 5.51349304415197470893758061348, 6.70592469241105562929820761127, 6.88272937685000054546244148602, 8.063775840951172586972464087393, 8.175715778102382297594997304705, 9.713899133429240960166209407847, 9.898035679329563907669108234156, 10.86343315491896409451241649589, 11.32089418591034566504780451062, 12.11294677755526756731381454911, 12.50043296467404237939544727902

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