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Properties

Label 4-18e2-1.1-c33e2-0-1
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $15417.9$
Root an. cond. $11.1431$
Motivic weight $33$
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  + 1.31e5·2-s + 1.28e10·4-s + 9.26e9·5-s − 3.73e13·7-s + 1.12e15·8-s + 1.21e15·10-s − 1.37e17·11-s + 2.55e18·13-s − 4.90e18·14-s + 9.22e19·16-s − 1.41e20·17-s − 7.60e20·19-s + 1.19e20·20-s − 1.80e22·22-s + 6.37e22·23-s − 1.94e23·25-s + 3.34e23·26-s − 4.81e23·28-s − 2.05e24·29-s + 7.09e23·31-s + 7.25e24·32-s − 1.85e25·34-s − 3.46e23·35-s + 8.09e25·37-s − 9.97e25·38-s + 1.04e25·40-s − 2.98e26·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.0271·5-s − 0.425·7-s + 1.41·8-s + 0.0384·10-s − 0.905·11-s + 1.06·13-s − 0.601·14-s + 5/4·16-s − 0.704·17-s − 0.605·19-s + 0.0407·20-s − 1.28·22-s + 2.16·23-s − 1.67·25-s + 1.50·26-s − 0.637·28-s − 1.52·29-s + 0.175·31-s + 1.06·32-s − 0.996·34-s − 0.0115·35-s + 1.07·37-s − 0.855·38-s + 0.0384·40-s − 0.730·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+33/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(15417.9\)
Root analytic conductor: \(11.1431\)
Motivic weight: \(33\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 324,\ (\ :33/2, 33/2),\ 1)\)

Particular Values

\(L(17)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{35}{2})\) 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$C_1$ \( ( 1 - p^{16} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 - 9266183796 T + 62398634444922367286 p^{5} T^{2} - 9266183796 p^{33} T^{3} + p^{66} T^{4} \)
7$D_{4}$ \( 1 + 37393677012848 T - \)\(20\!\cdots\!70\)\( p^{3} T^{2} + 37393677012848 p^{33} T^{3} + p^{66} T^{4} \)
11$D_{4}$ \( 1 + 12541024770321480 p T + \)\(32\!\cdots\!22\)\( p^{2} T^{2} + 12541024770321480 p^{34} T^{3} + p^{66} T^{4} \)
13$D_{4}$ \( 1 - 2553020813416682524 T + \)\(74\!\cdots\!50\)\( p^{2} T^{2} - 2553020813416682524 p^{33} T^{3} + p^{66} T^{4} \)
17$D_{4}$ \( 1 + \)\(14\!\cdots\!12\)\( T + \)\(26\!\cdots\!30\)\( p T^{2} + \)\(14\!\cdots\!12\)\( p^{33} T^{3} + p^{66} T^{4} \)
19$D_{4}$ \( 1 + \)\(76\!\cdots\!48\)\( T - \)\(85\!\cdots\!74\)\( p T^{2} + \)\(76\!\cdots\!48\)\( p^{33} T^{3} + p^{66} T^{4} \)
23$D_{4}$ \( 1 - \)\(63\!\cdots\!00\)\( T + \)\(11\!\cdots\!42\)\( p T^{2} - \)\(63\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \)
29$D_{4}$ \( 1 + \)\(20\!\cdots\!88\)\( T + \)\(10\!\cdots\!66\)\( p T^{2} + \)\(20\!\cdots\!88\)\( p^{33} T^{3} + p^{66} T^{4} \)
31$D_{4}$ \( 1 - \)\(70\!\cdots\!36\)\( T + \)\(83\!\cdots\!26\)\( p T^{2} - \)\(70\!\cdots\!36\)\( p^{33} T^{3} + p^{66} T^{4} \)
37$D_{4}$ \( 1 - \)\(80\!\cdots\!24\)\( T + \)\(12\!\cdots\!38\)\( T^{2} - \)\(80\!\cdots\!24\)\( p^{33} T^{3} + p^{66} T^{4} \)
41$D_{4}$ \( 1 + \)\(29\!\cdots\!68\)\( T + \)\(25\!\cdots\!98\)\( T^{2} + \)\(29\!\cdots\!68\)\( p^{33} T^{3} + p^{66} T^{4} \)
43$D_{4}$ \( 1 + \)\(49\!\cdots\!68\)\( T + \)\(11\!\cdots\!42\)\( T^{2} + \)\(49\!\cdots\!68\)\( p^{33} T^{3} + p^{66} T^{4} \)
47$D_{4}$ \( 1 + \)\(98\!\cdots\!40\)\( T + \)\(53\!\cdots\!54\)\( T^{2} + \)\(98\!\cdots\!40\)\( p^{33} T^{3} + p^{66} T^{4} \)
53$D_{4}$ \( 1 + \)\(49\!\cdots\!08\)\( T + \)\(22\!\cdots\!62\)\( T^{2} + \)\(49\!\cdots\!08\)\( p^{33} T^{3} + p^{66} T^{4} \)
59$D_{4}$ \( 1 + \)\(61\!\cdots\!80\)\( T + \)\(33\!\cdots\!58\)\( T^{2} + \)\(61\!\cdots\!80\)\( p^{33} T^{3} + p^{66} T^{4} \)
61$D_{4}$ \( 1 + \)\(48\!\cdots\!80\)\( T + \)\(16\!\cdots\!62\)\( T^{2} + \)\(48\!\cdots\!80\)\( p^{33} T^{3} + p^{66} T^{4} \)
67$D_{4}$ \( 1 + \)\(26\!\cdots\!68\)\( T + \)\(53\!\cdots\!30\)\( T^{2} + \)\(26\!\cdots\!68\)\( p^{33} T^{3} + p^{66} T^{4} \)
71$D_{4}$ \( 1 - \)\(31\!\cdots\!60\)\( T + \)\(15\!\cdots\!22\)\( T^{2} - \)\(31\!\cdots\!60\)\( p^{33} T^{3} + p^{66} T^{4} \)
73$D_{4}$ \( 1 + \)\(42\!\cdots\!56\)\( T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(42\!\cdots\!56\)\( p^{33} T^{3} + p^{66} T^{4} \)
79$D_{4}$ \( 1 - \)\(32\!\cdots\!96\)\( T + \)\(67\!\cdots\!82\)\( T^{2} - \)\(32\!\cdots\!96\)\( p^{33} T^{3} + p^{66} T^{4} \)
83$D_{4}$ \( 1 + \)\(14\!\cdots\!44\)\( T + \)\(93\!\cdots\!10\)\( T^{2} + \)\(14\!\cdots\!44\)\( p^{33} T^{3} + p^{66} T^{4} \)
89$D_{4}$ \( 1 + \)\(18\!\cdots\!56\)\( T + \)\(50\!\cdots\!22\)\( T^{2} + \)\(18\!\cdots\!56\)\( p^{33} T^{3} + p^{66} T^{4} \)
97$D_{4}$ \( 1 + \)\(26\!\cdots\!16\)\( T + \)\(65\!\cdots\!18\)\( T^{2} + \)\(26\!\cdots\!16\)\( p^{33} T^{3} + p^{66} T^{4} \)
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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

−11.62410784434982123216292652114, −11.22269371267109422516718122763, −10.82298582552851286060945143545, −10.04156773683521955321294026564, −9.293762260610593536873252024450, −8.569609824544656189883402240920, −7.76185945369932879254332428061, −7.28564343174753330013866821395, −6.29866733212744261017649521155, −6.25934464481657214747795393087, −5.38019238751031070989714881586, −4.87231240974929411528752381251, −4.22027165781374276573953914629, −3.62282345615777985866173920102, −2.98564684957662685629128880662, −2.63242383855454044815561364490, −1.57176699898742495641082025783, −1.47611311140320551809146860858, 0, 0, 1.47611311140320551809146860858, 1.57176699898742495641082025783, 2.63242383855454044815561364490, 2.98564684957662685629128880662, 3.62282345615777985866173920102, 4.22027165781374276573953914629, 4.87231240974929411528752381251, 5.38019238751031070989714881586, 6.25934464481657214747795393087, 6.29866733212744261017649521155, 7.28564343174753330013866821395, 7.76185945369932879254332428061, 8.569609824544656189883402240920, 9.293762260610593536873252024450, 10.04156773683521955321294026564, 10.82298582552851286060945143545, 11.22269371267109422516718122763, 11.62410784434982123216292652114

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