Properties

Label 4-18e2-1.1-c35e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $19508.0$
Root an. cond. $11.8182$
Motivic weight $35$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.62e5·2-s + 5.15e10·4-s + 1.44e12·5-s − 4.28e14·7-s + 9.00e15·8-s + 3.78e17·10-s + 9.22e17·11-s + 1.44e19·13-s − 1.12e20·14-s + 1.47e21·16-s + 7.38e21·17-s − 5.06e22·19-s + 7.44e22·20-s + 2.41e23·22-s + 3.31e23·23-s − 3.85e24·25-s + 3.77e24·26-s − 2.20e25·28-s + 7.91e25·29-s − 4.99e25·31-s + 2.32e26·32-s + 1.93e27·34-s − 6.19e26·35-s + 5.65e27·37-s − 1.32e28·38-s + 1.30e28·40-s − 4.92e27·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.846·5-s − 0.696·7-s + 1.41·8-s + 1.19·10-s + 0.550·11-s + 0.462·13-s − 0.984·14-s + 5/4·16-s + 2.16·17-s − 2.11·19-s + 1.27·20-s + 0.778·22-s + 0.490·23-s − 1.32·25-s + 0.653·26-s − 1.04·28-s + 2.02·29-s − 0.397·31-s + 1.06·32-s + 3.06·34-s − 0.589·35-s + 2.03·37-s − 2.99·38-s + 1.19·40-s − 0.294·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+35/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(19508.0\)
Root analytic conductor: \(11.8182\)
Motivic weight: \(35\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :35/2, 35/2),\ 1)\)

Particular Values

\(L(18)\) \(\approx\) \(10.60416503\)
\(L(\frac12)\) \(\approx\) \(10.60416503\)
\(L(\frac{37}{2})\) 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$C_1$ \( ( 1 - p^{17} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 - 1444437369204 T + \)\(95\!\cdots\!34\)\( p^{4} T^{2} - 1444437369204 p^{35} T^{3} + p^{70} T^{4} \)
7$D_{4}$ \( 1 + 61221064163936 p T - \)\(19\!\cdots\!38\)\( p^{4} T^{2} + 61221064163936 p^{36} T^{3} + p^{70} T^{4} \)
11$D_{4}$ \( 1 - 922360657028601576 T - \)\(74\!\cdots\!74\)\( p^{2} T^{2} - 922360657028601576 p^{35} T^{3} + p^{70} T^{4} \)
13$D_{4}$ \( 1 - 14414039247401633356 T - \)\(23\!\cdots\!58\)\( p^{2} T^{2} - 14414039247401633356 p^{35} T^{3} + p^{70} T^{4} \)
17$D_{4}$ \( 1 - \)\(73\!\cdots\!52\)\( T + \)\(21\!\cdots\!86\)\( p T^{2} - \)\(73\!\cdots\!52\)\( p^{35} T^{3} + p^{70} T^{4} \)
19$D_{4}$ \( 1 + \)\(26\!\cdots\!20\)\( p T + \)\(48\!\cdots\!18\)\( p^{2} T^{2} + \)\(26\!\cdots\!20\)\( p^{36} T^{3} + p^{70} T^{4} \)
23$D_{4}$ \( 1 - \)\(14\!\cdots\!68\)\( p T + \)\(17\!\cdots\!22\)\( p^{2} T^{2} - \)\(14\!\cdots\!68\)\( p^{36} T^{3} + p^{70} T^{4} \)
29$D_{4}$ \( 1 - \)\(79\!\cdots\!20\)\( T + \)\(13\!\cdots\!62\)\( p T^{2} - \)\(79\!\cdots\!20\)\( p^{35} T^{3} + p^{70} T^{4} \)
31$D_{4}$ \( 1 + \)\(16\!\cdots\!36\)\( p T + \)\(99\!\cdots\!06\)\( p^{2} T^{2} + \)\(16\!\cdots\!36\)\( p^{36} T^{3} + p^{70} T^{4} \)
37$D_{4}$ \( 1 - \)\(56\!\cdots\!48\)\( T + \)\(19\!\cdots\!62\)\( T^{2} - \)\(56\!\cdots\!48\)\( p^{35} T^{3} + p^{70} T^{4} \)
41$D_{4}$ \( 1 + \)\(49\!\cdots\!84\)\( T + \)\(48\!\cdots\!66\)\( T^{2} + \)\(49\!\cdots\!84\)\( p^{35} T^{3} + p^{70} T^{4} \)
43$D_{4}$ \( 1 - \)\(22\!\cdots\!76\)\( T + \)\(29\!\cdots\!58\)\( T^{2} - \)\(22\!\cdots\!76\)\( p^{35} T^{3} + p^{70} T^{4} \)
47$D_{4}$ \( 1 + \)\(16\!\cdots\!88\)\( T + \)\(26\!\cdots\!22\)\( T^{2} + \)\(16\!\cdots\!88\)\( p^{35} T^{3} + p^{70} T^{4} \)
53$D_{4}$ \( 1 + \)\(40\!\cdots\!36\)\( T + \)\(81\!\cdots\!38\)\( T^{2} + \)\(40\!\cdots\!36\)\( p^{35} T^{3} + p^{70} T^{4} \)
59$D_{4}$ \( 1 + \)\(15\!\cdots\!80\)\( T + \)\(23\!\cdots\!98\)\( T^{2} + \)\(15\!\cdots\!80\)\( p^{35} T^{3} + p^{70} T^{4} \)
61$D_{4}$ \( 1 - \)\(14\!\cdots\!24\)\( T + \)\(54\!\cdots\!46\)\( T^{2} - \)\(14\!\cdots\!24\)\( p^{35} T^{3} + p^{70} T^{4} \)
67$D_{4}$ \( 1 + \)\(11\!\cdots\!12\)\( T + \)\(18\!\cdots\!22\)\( T^{2} + \)\(11\!\cdots\!12\)\( p^{35} T^{3} + p^{70} T^{4} \)
71$D_{4}$ \( 1 - \)\(27\!\cdots\!76\)\( T + \)\(11\!\cdots\!46\)\( T^{2} - \)\(27\!\cdots\!76\)\( p^{35} T^{3} + p^{70} T^{4} \)
73$D_{4}$ \( 1 - \)\(45\!\cdots\!36\)\( T + \)\(35\!\cdots\!38\)\( T^{2} - \)\(45\!\cdots\!36\)\( p^{35} T^{3} + p^{70} T^{4} \)
79$D_{4}$ \( 1 + \)\(23\!\cdots\!20\)\( T + \)\(65\!\cdots\!98\)\( T^{2} + \)\(23\!\cdots\!20\)\( p^{35} T^{3} + p^{70} T^{4} \)
83$D_{4}$ \( 1 - \)\(70\!\cdots\!44\)\( T + \)\(28\!\cdots\!98\)\( T^{2} - \)\(70\!\cdots\!44\)\( p^{35} T^{3} + p^{70} T^{4} \)
89$D_{4}$ \( 1 + \)\(18\!\cdots\!20\)\( T + \)\(38\!\cdots\!98\)\( T^{2} + \)\(18\!\cdots\!20\)\( p^{35} T^{3} + p^{70} T^{4} \)
97$D_{4}$ \( 1 - \)\(64\!\cdots\!88\)\( T + \)\(78\!\cdots\!22\)\( T^{2} - \)\(64\!\cdots\!88\)\( p^{35} T^{3} + p^{70} T^{4} \)
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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

−12.26956248041553918113600996812, −11.71929481023437031875902116701, −10.90661875225559534975776841129, −10.41218867726460893510092310562, −9.720746632345450072262228706854, −9.346516217812908056022842646880, −8.137839987134956734439162954038, −7.898533900498926792484424944230, −6.67886441014351257202085415517, −6.51027709037044531364634146340, −5.75144654848116017169080937180, −5.70888744381462452446968944504, −4.43301847727627941413995005944, −4.39337802430081869577012648295, −3.40705038835848556099890728039, −3.05105798242150536102343079291, −2.43901980970826778346307523730, −1.65156411514006338061780290700, −1.30727406435824235506017535930, −0.47212513305332821456808969803, 0.47212513305332821456808969803, 1.30727406435824235506017535930, 1.65156411514006338061780290700, 2.43901980970826778346307523730, 3.05105798242150536102343079291, 3.40705038835848556099890728039, 4.39337802430081869577012648295, 4.43301847727627941413995005944, 5.70888744381462452446968944504, 5.75144654848116017169080937180, 6.51027709037044531364634146340, 6.67886441014351257202085415517, 7.898533900498926792484424944230, 8.137839987134956734439162954038, 9.346516217812908056022842646880, 9.720746632345450072262228706854, 10.41218867726460893510092310562, 10.90661875225559534975776841129, 11.71929481023437031875902116701, 12.26956248041553918113600996812

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