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Properties

Label 4-18e2-1.1-c25e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $5080.75$
Root an. cond. $8.44271$
Motivic weight $25$
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  − 8.19e3·2-s + 5.03e7·4-s − 7.41e8·5-s − 3.76e8·7-s − 2.74e11·8-s + 6.07e12·10-s − 8.32e12·11-s − 1.06e14·13-s + 3.08e12·14-s + 1.40e15·16-s − 1.32e15·17-s − 4.77e14·19-s − 3.73e16·20-s + 6.81e16·22-s + 1.15e17·23-s + 7.49e16·25-s + 8.72e17·26-s − 1.89e16·28-s − 1.72e18·29-s − 8.68e18·31-s − 6.91e18·32-s + 1.08e19·34-s + 2.79e17·35-s − 3.49e19·37-s + 3.90e18·38-s + 2.03e20·40-s − 8.30e19·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.35·5-s − 0.0102·7-s − 1.41·8-s + 1.92·10-s − 0.799·11-s − 1.26·13-s + 0.0145·14-s + 5/4·16-s − 0.552·17-s − 0.0494·19-s − 2.03·20-s + 1.13·22-s + 1.09·23-s + 0.251·25-s + 1.79·26-s − 0.0154·28-s − 0.905·29-s − 1.98·31-s − 1.06·32-s + 0.781·34-s + 0.0139·35-s − 0.871·37-s + 0.0699·38-s + 1.92·40-s − 0.574·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(5080.75\)
Root analytic conductor: \(8.44271\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(0.5693650632\)
\(L(\frac12)\) \(\approx\) \(0.5693650632\)
\(L(\frac{27}{2})\) 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$C_1$ \( ( 1 + p^{12} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 + 29678124 p^{2} T + 760941305600974 p^{4} T^{2} + 29678124 p^{27} T^{3} + p^{50} T^{4} \)
7$D_{4}$ \( 1 + 53790992 p T + 736567609831341198 p^{4} T^{2} + 53790992 p^{26} T^{3} + p^{50} T^{4} \)
11$D_{4}$ \( 1 + 756639510024 p T + \)\(17\!\cdots\!46\)\( p^{3} T^{2} + 756639510024 p^{26} T^{3} + p^{50} T^{4} \)
13$D_{4}$ \( 1 + 106467053152292 T + \)\(11\!\cdots\!54\)\( p T^{2} + 106467053152292 p^{25} T^{3} + p^{50} T^{4} \)
17$D_{4}$ \( 1 + 1327878920113956 T + \)\(39\!\cdots\!94\)\( p T^{2} + 1327878920113956 p^{25} T^{3} + p^{50} T^{4} \)
19$D_{4}$ \( 1 + 477079242949400 T + \)\(83\!\cdots\!42\)\( p T^{2} + 477079242949400 p^{25} T^{3} + p^{50} T^{4} \)
23$D_{4}$ \( 1 - 5013261990498864 p T + \)\(43\!\cdots\!58\)\( p^{2} T^{2} - 5013261990498864 p^{26} T^{3} + p^{50} T^{4} \)
29$D_{4}$ \( 1 + 1724412645206435580 T + \)\(30\!\cdots\!98\)\( T^{2} + 1724412645206435580 p^{25} T^{3} + p^{50} T^{4} \)
31$D_{4}$ \( 1 + 8688082288351126976 T + \)\(57\!\cdots\!46\)\( T^{2} + 8688082288351126976 p^{25} T^{3} + p^{50} T^{4} \)
37$D_{4}$ \( 1 + 34908364049750170484 T + \)\(27\!\cdots\!78\)\( T^{2} + 34908364049750170484 p^{25} T^{3} + p^{50} T^{4} \)
41$D_{4}$ \( 1 + 83014324355953468884 T + \)\(29\!\cdots\!66\)\( T^{2} + 83014324355953468884 p^{25} T^{3} + p^{50} T^{4} \)
43$D_{4}$ \( 1 - 44539608583471901848 T + \)\(11\!\cdots\!62\)\( T^{2} - 44539608583471901848 p^{25} T^{3} + p^{50} T^{4} \)
47$D_{4}$ \( 1 + 39773632596907970208 p T + \)\(19\!\cdots\!58\)\( T^{2} + 39773632596907970208 p^{26} T^{3} + p^{50} T^{4} \)
53$D_{4}$ \( 1 - \)\(32\!\cdots\!32\)\( T + \)\(28\!\cdots\!42\)\( T^{2} - \)\(32\!\cdots\!32\)\( p^{25} T^{3} + p^{50} T^{4} \)
59$D_{4}$ \( 1 - \)\(17\!\cdots\!40\)\( T + \)\(42\!\cdots\!98\)\( T^{2} - \)\(17\!\cdots\!40\)\( p^{25} T^{3} + p^{50} T^{4} \)
61$D_{4}$ \( 1 - \)\(33\!\cdots\!44\)\( T + \)\(85\!\cdots\!86\)\( T^{2} - \)\(33\!\cdots\!44\)\( p^{25} T^{3} + p^{50} T^{4} \)
67$D_{4}$ \( 1 - \)\(33\!\cdots\!16\)\( T + \)\(43\!\cdots\!78\)\( T^{2} - \)\(33\!\cdots\!16\)\( p^{25} T^{3} + p^{50} T^{4} \)
71$D_{4}$ \( 1 - \)\(27\!\cdots\!56\)\( T + \)\(49\!\cdots\!86\)\( T^{2} - \)\(27\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} \)
73$D_{4}$ \( 1 - \)\(31\!\cdots\!48\)\( T + \)\(86\!\cdots\!62\)\( T^{2} - \)\(31\!\cdots\!48\)\( p^{25} T^{3} + p^{50} T^{4} \)
79$D_{4}$ \( 1 - \)\(92\!\cdots\!20\)\( T + \)\(73\!\cdots\!98\)\( T^{2} - \)\(92\!\cdots\!20\)\( p^{25} T^{3} + p^{50} T^{4} \)
83$D_{4}$ \( 1 - \)\(45\!\cdots\!52\)\( T - \)\(76\!\cdots\!38\)\( T^{2} - \)\(45\!\cdots\!52\)\( p^{25} T^{3} + p^{50} T^{4} \)
89$D_{4}$ \( 1 - \)\(23\!\cdots\!20\)\( T + \)\(80\!\cdots\!98\)\( T^{2} - \)\(23\!\cdots\!20\)\( p^{25} T^{3} + p^{50} T^{4} \)
97$D_{4}$ \( 1 - \)\(13\!\cdots\!36\)\( T + \)\(12\!\cdots\!38\)\( T^{2} - \)\(13\!\cdots\!36\)\( p^{25} T^{3} + p^{50} T^{4} \)
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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

−12.96154789154516291696789308609, −12.83013168040266748110854379406, −11.76747852508582952478656255176, −11.43309906837019624662161005762, −10.86376683054453447314147730428, −10.16480568573335293048442514129, −9.462021871151880350361058659983, −8.911294944649996145230102612745, −7.946161488155440123082930209399, −7.894258507363011768835935329992, −7.01820866281378949982093163759, −6.70512306033698209449863210976, −5.30424348819325161259777202821, −4.94884701779025338058325453859, −3.63730450553272444843802071264, −3.35808716527520320718013534778, −2.10503367000393694134246411881, −2.01854088583459914493392596188, −0.56008546852882636750370691766, −0.42036090688098676328035739724, 0.42036090688098676328035739724, 0.56008546852882636750370691766, 2.01854088583459914493392596188, 2.10503367000393694134246411881, 3.35808716527520320718013534778, 3.63730450553272444843802071264, 4.94884701779025338058325453859, 5.30424348819325161259777202821, 6.70512306033698209449863210976, 7.01820866281378949982093163759, 7.894258507363011768835935329992, 7.946161488155440123082930209399, 8.911294944649996145230102612745, 9.462021871151880350361058659983, 10.16480568573335293048442514129, 10.86376683054453447314147730428, 11.43309906837019624662161005762, 11.76747852508582952478656255176, 12.83013168040266748110854379406, 12.96154789154516291696789308609

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