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Properties

Label 4-18e2-1.1-c23e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $3640.52$
Root an. cond. $7.76767$
Motivic weight $23$
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  − 4.09e3·2-s + 1.25e7·4-s + 3.11e7·5-s − 4.70e9·7-s − 3.43e10·8-s − 1.27e11·10-s − 5.26e11·11-s + 2.59e12·13-s + 1.92e13·14-s + 8.79e13·16-s + 2.24e14·17-s − 3.91e14·19-s + 3.91e14·20-s + 2.15e15·22-s + 1.45e15·23-s − 1.73e16·25-s − 1.06e16·26-s − 5.92e16·28-s − 5.55e16·29-s − 1.35e17·31-s − 2.16e17·32-s − 9.20e17·34-s − 1.46e17·35-s − 2.80e18·37-s + 1.60e18·38-s − 1.06e18·40-s − 1.91e18·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.285·5-s − 0.899·7-s − 1.41·8-s − 0.403·10-s − 0.556·11-s + 0.401·13-s + 1.27·14-s + 5/4·16-s + 1.59·17-s − 0.771·19-s + 0.427·20-s + 0.787·22-s + 0.318·23-s − 1.45·25-s − 0.568·26-s − 1.34·28-s − 0.845·29-s − 0.958·31-s − 1.06·32-s − 2.24·34-s − 0.256·35-s − 2.59·37-s + 1.09·38-s − 0.403·40-s − 0.542·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3640.52\)
Root analytic conductor: \(7.76767\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(0.1879290248\)
\(L(\frac12)\) \(\approx\) \(0.1879290248\)
\(L(\frac{25}{2})\) 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$C_1$ \( ( 1 + p^{11} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 - 6228096 p T + 29246094570106 p^{4} T^{2} - 6228096 p^{24} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 + 4706401400 T + 588305895824182110 p^{2} T^{2} + 4706401400 p^{23} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 + 47906695680 p T - \)\(11\!\cdots\!14\)\( p^{2} T^{2} + 47906695680 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 - 2596032576340 T + \)\(16\!\cdots\!30\)\( p T^{2} - 2596032576340 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 - 13218608454912 p T + \)\(18\!\cdots\!70\)\( p^{2} T^{2} - 13218608454912 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 + 391793415257552 T + \)\(25\!\cdots\!26\)\( p T^{2} + 391793415257552 p^{23} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 - 1457565951360000 T + \)\(41\!\cdots\!34\)\( T^{2} - 1457565951360000 p^{23} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 + 55565590118160000 T + \)\(72\!\cdots\!02\)\( T^{2} + 55565590118160000 p^{23} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 + 135626244868824248 T + \)\(43\!\cdots\!58\)\( T^{2} + 135626244868824248 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 + 2809007989962930500 T + \)\(40\!\cdots\!90\)\( T^{2} + 2809007989962930500 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 + 46604458931600640 p T + \)\(24\!\cdots\!42\)\( T^{2} + 46604458931600640 p^{24} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 + 7181641338989011280 T + \)\(44\!\cdots\!78\)\( T^{2} + 7181641338989011280 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 - 7006081021857254400 T + \)\(21\!\cdots\!46\)\( T^{2} - 7006081021857254400 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 - \)\(11\!\cdots\!36\)\( T + \)\(60\!\cdots\!78\)\( T^{2} - \)\(11\!\cdots\!36\)\( p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 - \)\(50\!\cdots\!40\)\( T + \)\(15\!\cdots\!42\)\( T^{2} - \)\(50\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 - 50083670456482700044 T + \)\(12\!\cdots\!46\)\( T^{2} - 50083670456482700044 p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 + \)\(71\!\cdots\!00\)\( T + \)\(72\!\cdots\!02\)\( T^{2} + \)\(71\!\cdots\!00\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 - \)\(69\!\cdots\!40\)\( T + \)\(19\!\cdots\!18\)\( T^{2} - \)\(69\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 + \)\(35\!\cdots\!00\)\( T + \)\(13\!\cdots\!34\)\( T^{2} + \)\(35\!\cdots\!00\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 - \)\(20\!\cdots\!68\)\( T + \)\(77\!\cdots\!34\)\( T^{2} - \)\(20\!\cdots\!68\)\( p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 - \)\(14\!\cdots\!08\)\( T + \)\(90\!\cdots\!90\)\( T^{2} - \)\(14\!\cdots\!08\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 - \)\(37\!\cdots\!80\)\( T + \)\(16\!\cdots\!14\)\( T^{2} - \)\(37\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 - \)\(18\!\cdots\!20\)\( T + \)\(99\!\cdots\!10\)\( T^{2} - \)\(18\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−13.58421921848767519870240234562, −13.15540181334259978184329789096, −12.18454759112620994193846971549, −11.88215426710114872839141465731, −10.71789789037948043521158531706, −10.49640417256712904273975142895, −9.666804538292304715861303007073, −9.405777164102424881223949416122, −8.381846011626154471609648534080, −8.071296624828015445304511930279, −7.02799478915903212023780054962, −6.76035075165845655630007208593, −5.56779646153068732642364286584, −5.47451056826968467610188933799, −3.63942576695126561392445817361, −3.47734756677127518863981930149, −2.32545138210439441656630909973, −1.83349695022806916460993457242, −1.03349682517827863338034908301, −0.15367982818077552883609015158, 0.15367982818077552883609015158, 1.03349682517827863338034908301, 1.83349695022806916460993457242, 2.32545138210439441656630909973, 3.47734756677127518863981930149, 3.63942576695126561392445817361, 5.47451056826968467610188933799, 5.56779646153068732642364286584, 6.76035075165845655630007208593, 7.02799478915903212023780054962, 8.071296624828015445304511930279, 8.381846011626154471609648534080, 9.405777164102424881223949416122, 9.666804538292304715861303007073, 10.49640417256712904273975142895, 10.71789789037948043521158531706, 11.88215426710114872839141465731, 12.18454759112620994193846971549, 13.15540181334259978184329789096, 13.58421921848767519870240234562

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