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Properties

Label 4-18e2-1.1-c21e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $2530.68$
Root an. cond. $7.09266$
Motivic weight $21$
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  + 2.04e3·2-s + 3.14e6·4-s + 1.52e7·5-s + 5.47e8·7-s + 4.29e9·8-s + 3.12e10·10-s + 5.05e10·11-s + 2.16e11·13-s + 1.12e12·14-s + 5.49e12·16-s + 9.47e12·17-s − 1.44e13·19-s + 4.79e13·20-s + 1.03e14·22-s − 3.12e14·23-s + 1.64e14·25-s + 4.43e14·26-s + 1.72e15·28-s + 8.49e14·29-s + 8.82e15·31-s + 6.75e15·32-s + 1.94e16·34-s + 8.34e15·35-s + 3.28e16·37-s − 2.95e16·38-s + 6.54e16·40-s + 4.05e16·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.698·5-s + 0.732·7-s + 1.41·8-s + 0.987·10-s + 0.587·11-s + 0.435·13-s + 1.03·14-s + 5/4·16-s + 1.13·17-s − 0.540·19-s + 1.04·20-s + 0.830·22-s − 1.57·23-s + 0.344·25-s + 0.616·26-s + 1.09·28-s + 0.374·29-s + 1.93·31-s + 1.06·32-s + 1.61·34-s + 0.511·35-s + 1.12·37-s − 0.763·38-s + 0.987·40-s + 0.471·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2530.68\)
Root analytic conductor: \(7.09266\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(14.63990850\)
\(L(\frac12)\) \(\approx\) \(14.63990850\)
\(L(\frac{23}{2})\) 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$C_1$ \( ( 1 - p^{10} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 - 15247776 T + 2722535744122 p^{2} T^{2} - 15247776 p^{21} T^{3} + p^{42} T^{4} \)
7$D_{4}$ \( 1 - 78231400 p T + 24019670123427390 p^{2} T^{2} - 78231400 p^{22} T^{3} + p^{42} T^{4} \)
11$D_{4}$ \( 1 - 50505118080 T + \)\(14\!\cdots\!06\)\( p T^{2} - 50505118080 p^{21} T^{3} + p^{42} T^{4} \)
13$D_{4}$ \( 1 - 216579607780 T - \)\(53\!\cdots\!70\)\( p T^{2} - 216579607780 p^{21} T^{3} + p^{42} T^{4} \)
17$D_{4}$ \( 1 - 9473317420608 T + \)\(89\!\cdots\!50\)\( p T^{2} - 9473317420608 p^{21} T^{3} + p^{42} T^{4} \)
19$D_{4}$ \( 1 + 759692409872 p T + \)\(16\!\cdots\!54\)\( p^{2} T^{2} + 759692409872 p^{22} T^{3} + p^{42} T^{4} \)
23$D_{4}$ \( 1 + 312567491546880 T + \)\(63\!\cdots\!46\)\( T^{2} + 312567491546880 p^{21} T^{3} + p^{42} T^{4} \)
29$D_{4}$ \( 1 - 849424935919200 T + \)\(83\!\cdots\!22\)\( T^{2} - 849424935919200 p^{21} T^{3} + p^{42} T^{4} \)
31$D_{4}$ \( 1 - 8822955019110232 T + \)\(44\!\cdots\!18\)\( T^{2} - 8822955019110232 p^{21} T^{3} + p^{42} T^{4} \)
37$D_{4}$ \( 1 - 32850121839364300 T + \)\(14\!\cdots\!10\)\( T^{2} - 32850121839364300 p^{21} T^{3} + p^{42} T^{4} \)
41$D_{4}$ \( 1 - 40560129801992640 T + \)\(12\!\cdots\!82\)\( T^{2} - 40560129801992640 p^{21} T^{3} + p^{42} T^{4} \)
43$D_{4}$ \( 1 + 131779045482586160 T + \)\(38\!\cdots\!42\)\( T^{2} + 131779045482586160 p^{21} T^{3} + p^{42} T^{4} \)
47$D_{4}$ \( 1 - 815264473551648000 T + \)\(38\!\cdots\!94\)\( T^{2} - 815264473551648000 p^{21} T^{3} + p^{42} T^{4} \)
53$D_{4}$ \( 1 - 2534003124574282848 T + \)\(45\!\cdots\!82\)\( T^{2} - 2534003124574282848 p^{21} T^{3} + p^{42} T^{4} \)
59$D_{4}$ \( 1 - 10926573350713847040 T + \)\(59\!\cdots\!22\)\( T^{2} - 10926573350713847040 p^{21} T^{3} + p^{42} T^{4} \)
61$D_{4}$ \( 1 - 4128039719817921724 T + \)\(47\!\cdots\!66\)\( T^{2} - 4128039719817921724 p^{21} T^{3} + p^{42} T^{4} \)
67$D_{4}$ \( 1 - 25702995334270856800 T + \)\(52\!\cdots\!18\)\( T^{2} - 25702995334270856800 p^{21} T^{3} + p^{42} T^{4} \)
71$D_{4}$ \( 1 - 49254169402310261760 T + \)\(17\!\cdots\!78\)\( T^{2} - 49254169402310261760 p^{21} T^{3} + p^{42} T^{4} \)
73$D_{4}$ \( 1 + 56277128109279650900 T + \)\(34\!\cdots\!46\)\( T^{2} + 56277128109279650900 p^{21} T^{3} + p^{42} T^{4} \)
79$D_{4}$ \( 1 + \)\(15\!\cdots\!88\)\( T + \)\(13\!\cdots\!94\)\( T^{2} + \)\(15\!\cdots\!88\)\( p^{21} T^{3} + p^{42} T^{4} \)
83$D_{4}$ \( 1 + \)\(11\!\cdots\!76\)\( T + \)\(31\!\cdots\!10\)\( T^{2} + \)\(11\!\cdots\!76\)\( p^{21} T^{3} + p^{42} T^{4} \)
89$D_{4}$ \( 1 + \)\(19\!\cdots\!20\)\( T - \)\(11\!\cdots\!06\)\( T^{2} + \)\(19\!\cdots\!20\)\( p^{21} T^{3} + p^{42} T^{4} \)
97$D_{4}$ \( 1 + \)\(27\!\cdots\!40\)\( T + \)\(59\!\cdots\!10\)\( T^{2} + \)\(27\!\cdots\!40\)\( p^{21} T^{3} + p^{42} T^{4} \)
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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

−13.93875430535513829966495316442, −13.77135273344331512088435555356, −12.89982451077399111544553868184, −12.25741495832191779790845501056, −11.66362948616618656757154485026, −11.18638400099418859673108586912, −10.01314218848181286575374721356, −9.998653972122685201847368251179, −8.472898457324036238824978133659, −8.087884135252826269853838922160, −7.00106278220939281100399438920, −6.37198586397887354992376773090, −5.69238684569831684898408582332, −5.23148524713679476916546029150, −4.07808608529976694570339931205, −4.03249589438924083252620758703, −2.73260577028631809352545820965, −2.28015765268853862252857065646, −1.36851732650894502332431418760, −0.857995137995172223471328817637, 0.857995137995172223471328817637, 1.36851732650894502332431418760, 2.28015765268853862252857065646, 2.73260577028631809352545820965, 4.03249589438924083252620758703, 4.07808608529976694570339931205, 5.23148524713679476916546029150, 5.69238684569831684898408582332, 6.37198586397887354992376773090, 7.00106278220939281100399438920, 8.087884135252826269853838922160, 8.472898457324036238824978133659, 9.998653972122685201847368251179, 10.01314218848181286575374721356, 11.18638400099418859673108586912, 11.66362948616618656757154485026, 12.25741495832191779790845501056, 12.89982451077399111544553868184, 13.77135273344331512088435555356, 13.93875430535513829966495316442

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