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Properties

Label 4-18e2-1.1-c43e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $44436.0$
Root an. cond. $14.5189$
Motivic weight $43$
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  + 4.19e6·2-s + 1.31e13·4-s + 3.98e14·5-s + 1.17e18·7-s + 3.68e19·8-s + 1.67e21·10-s + 1.16e22·11-s − 1.65e24·13-s + 4.92e24·14-s + 9.67e25·16-s − 3.43e26·17-s − 5.26e26·19-s + 5.26e27·20-s + 4.87e28·22-s − 2.29e29·23-s + 1.34e30·25-s − 6.95e30·26-s + 1.55e31·28-s − 3.41e31·29-s + 1.62e32·31-s + 2.43e32·32-s − 1.43e33·34-s + 4.68e32·35-s − 4.39e33·37-s − 2.20e33·38-s + 1.47e34·40-s − 7.32e34·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.373·5-s + 0.795·7-s + 1.41·8-s + 0.528·10-s + 0.473·11-s − 1.86·13-s + 1.12·14-s + 5/4·16-s − 1.20·17-s − 0.169·19-s + 0.560·20-s + 0.669·22-s − 1.21·23-s + 1.18·25-s − 2.63·26-s + 1.19·28-s − 1.23·29-s + 1.39·31-s + 1.06·32-s − 1.70·34-s + 0.297·35-s − 0.843·37-s − 0.239·38-s + 0.528·40-s − 1.54·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+43/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(44436.0\)
Root analytic conductor: \(14.5189\)
Motivic weight: \(43\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :43/2, 43/2),\ 1)\)

Particular Values

\(L(22)\) \(\approx\) \(0.1006057016\)
\(L(\frac12)\) \(\approx\) \(0.1006057016\)
\(L(\frac{45}{2})\) 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$C_1$ \( ( 1 - p^{21} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 - 637860338052 p^{4} T - \)\(24\!\cdots\!58\)\( p^{11} T^{2} - 637860338052 p^{47} T^{3} + p^{86} T^{4} \)
7$D_{4}$ \( 1 - 1174870033543241008 T + \)\(91\!\cdots\!14\)\( p^{3} T^{2} - 1174870033543241008 p^{43} T^{3} + p^{86} T^{4} \)
11$D_{4}$ \( 1 - \)\(10\!\cdots\!16\)\( p T + \)\(32\!\cdots\!86\)\( p^{2} T^{2} - \)\(10\!\cdots\!16\)\( p^{44} T^{3} + p^{86} T^{4} \)
13$D_{4}$ \( 1 + \)\(12\!\cdots\!68\)\( p T + \)\(74\!\cdots\!78\)\( p^{4} T^{2} + \)\(12\!\cdots\!68\)\( p^{44} T^{3} + p^{86} T^{4} \)
17$D_{4}$ \( 1 + \)\(20\!\cdots\!04\)\( p T + \)\(32\!\cdots\!14\)\( p^{3} T^{2} + \)\(20\!\cdots\!04\)\( p^{44} T^{3} + p^{86} T^{4} \)
19$D_{4}$ \( 1 + \)\(52\!\cdots\!00\)\( T + \)\(65\!\cdots\!22\)\( p T^{2} + \)\(52\!\cdots\!00\)\( p^{43} T^{3} + p^{86} T^{4} \)
23$D_{4}$ \( 1 + \)\(22\!\cdots\!36\)\( T + \)\(31\!\cdots\!46\)\( p T^{2} + \)\(22\!\cdots\!36\)\( p^{43} T^{3} + p^{86} T^{4} \)
29$D_{4}$ \( 1 + \)\(11\!\cdots\!80\)\( p T + \)\(19\!\cdots\!58\)\( p^{2} T^{2} + \)\(11\!\cdots\!80\)\( p^{44} T^{3} + p^{86} T^{4} \)
31$D_{4}$ \( 1 - \)\(16\!\cdots\!04\)\( T + \)\(10\!\cdots\!06\)\( p T^{2} - \)\(16\!\cdots\!04\)\( p^{43} T^{3} + p^{86} T^{4} \)
37$D_{4}$ \( 1 + \)\(11\!\cdots\!36\)\( p T + \)\(21\!\cdots\!98\)\( p^{2} T^{2} + \)\(11\!\cdots\!36\)\( p^{44} T^{3} + p^{86} T^{4} \)
41$D_{4}$ \( 1 + \)\(17\!\cdots\!64\)\( p T + \)\(34\!\cdots\!06\)\( p^{2} T^{2} + \)\(17\!\cdots\!64\)\( p^{44} T^{3} + p^{86} T^{4} \)
43$D_{4}$ \( 1 - \)\(23\!\cdots\!36\)\( T + \)\(41\!\cdots\!38\)\( T^{2} - \)\(23\!\cdots\!36\)\( p^{43} T^{3} + p^{86} T^{4} \)
47$D_{4}$ \( 1 - \)\(39\!\cdots\!92\)\( T + \)\(27\!\cdots\!62\)\( T^{2} - \)\(39\!\cdots\!92\)\( p^{43} T^{3} + p^{86} T^{4} \)
53$D_{4}$ \( 1 + \)\(21\!\cdots\!56\)\( T + \)\(24\!\cdots\!38\)\( T^{2} + \)\(21\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \)
59$D_{4}$ \( 1 + \)\(67\!\cdots\!40\)\( T + \)\(20\!\cdots\!58\)\( T^{2} + \)\(67\!\cdots\!40\)\( p^{43} T^{3} + p^{86} T^{4} \)
61$D_{4}$ \( 1 - \)\(10\!\cdots\!04\)\( T + \)\(94\!\cdots\!66\)\( T^{2} - \)\(10\!\cdots\!04\)\( p^{43} T^{3} + p^{86} T^{4} \)
67$D_{4}$ \( 1 + \)\(12\!\cdots\!52\)\( T + \)\(36\!\cdots\!02\)\( T^{2} + \)\(12\!\cdots\!52\)\( p^{43} T^{3} + p^{86} T^{4} \)
71$D_{4}$ \( 1 + \)\(10\!\cdots\!64\)\( T + \)\(10\!\cdots\!46\)\( T^{2} + \)\(10\!\cdots\!64\)\( p^{43} T^{3} + p^{86} T^{4} \)
73$D_{4}$ \( 1 + \)\(10\!\cdots\!24\)\( T + \)\(26\!\cdots\!78\)\( T^{2} + \)\(10\!\cdots\!24\)\( p^{43} T^{3} + p^{86} T^{4} \)
79$D_{4}$ \( 1 + \)\(66\!\cdots\!20\)\( T + \)\(90\!\cdots\!78\)\( T^{2} + \)\(66\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \)
83$D_{4}$ \( 1 + \)\(54\!\cdots\!56\)\( T + \)\(14\!\cdots\!58\)\( T^{2} + \)\(54\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \)
89$D_{4}$ \( 1 + \)\(29\!\cdots\!20\)\( T + \)\(85\!\cdots\!38\)\( T^{2} + \)\(29\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \)
97$D_{4}$ \( 1 + \)\(22\!\cdots\!12\)\( T + \)\(34\!\cdots\!82\)\( T^{2} + \)\(22\!\cdots\!12\)\( p^{43} T^{3} + p^{86} T^{4} \)
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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

−11.25373387108486598027854587258, −11.09972577193908961473235328162, −10.11547373927905715145769122114, −9.867092935316109332709227606833, −8.920911022933908613622959677258, −8.390064593646455087225946579151, −7.48543238262455025756069486947, −7.26443273920034755539839816096, −6.48846178036149574743316425531, −6.11018745300065146018897768912, −5.33554539310354800165239160049, −4.91605924000051786744173197148, −4.34924712301280247466979327979, −4.20599150388047417253460560803, −3.10230786323827803582753876150, −2.78782277555394359080097902325, −2.12093992964695140646221973897, −1.71390434712692836253413348402, −1.26696615482664110575037103753, −0.03768311373459511149495553151, 0.03768311373459511149495553151, 1.26696615482664110575037103753, 1.71390434712692836253413348402, 2.12093992964695140646221973897, 2.78782277555394359080097902325, 3.10230786323827803582753876150, 4.20599150388047417253460560803, 4.34924712301280247466979327979, 4.91605924000051786744173197148, 5.33554539310354800165239160049, 6.11018745300065146018897768912, 6.48846178036149574743316425531, 7.26443273920034755539839816096, 7.48543238262455025756069486947, 8.390064593646455087225946579151, 8.920911022933908613622959677258, 9.867092935316109332709227606833, 10.11547373927905715145769122114, 11.09972577193908961473235328162, 11.25373387108486598027854587258

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