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Properties

Label 4-3e2-1.1-c23e2-0-0
Degree $4$
Conductor $9$
Sign $1$
Analytic cond. $101.125$
Root an. cond. $3.17113$
Motivic weight $23$
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  − 1.24e3·2-s − 3.54e5·3-s − 1.08e7·4-s − 4.68e7·5-s + 4.40e8·6-s − 2.11e8·7-s + 1.84e10·8-s + 9.41e10·9-s + 5.81e10·10-s + 1.46e12·11-s + 3.84e12·12-s + 1.04e13·13-s + 2.63e11·14-s + 1.65e13·15-s + 5.40e13·16-s − 2.10e14·17-s − 1.16e14·18-s − 9.07e14·19-s + 5.07e14·20-s + 7.50e13·21-s − 1.82e15·22-s + 1.01e16·23-s − 6.53e15·24-s − 3.42e15·25-s − 1.30e16·26-s − 2.22e16·27-s + 2.29e15·28-s + ⋯
L(s)  = 1  − 0.428·2-s − 1.15·3-s − 1.29·4-s − 0.428·5-s + 0.495·6-s − 0.0405·7-s + 0.758·8-s + 9-s + 0.183·10-s + 1.55·11-s + 1.49·12-s + 1.62·13-s + 0.0173·14-s + 0.495·15-s + 0.768·16-s − 1.49·17-s − 0.428·18-s − 1.78·19-s + 0.554·20-s + 0.0467·21-s − 0.665·22-s + 2.21·23-s − 0.876·24-s − 0.286·25-s − 0.696·26-s − 0.769·27-s + 0.0523·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(101.125\)
Root analytic conductor: \(3.17113\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(0.5414558829\)
\(L(\frac12)\) \(\approx\) \(0.5414558829\)
\(L(\frac{25}{2})\) 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)$
bad3$C_1$ \( ( 1 + p^{11} T )^{2} \)
good2$D_{4}$ \( 1 + 621 p T + 96791 p^{7} T^{2} + 621 p^{24} T^{3} + p^{46} T^{4} \)
5$D_{4}$ \( 1 + 9361764 p T + 8977934193886 p^{4} T^{2} + 9361764 p^{24} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 + 211963904 T - 175324586455056306 p^{2} T^{2} + 211963904 p^{23} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 - 133542942408 p T + \)\(18\!\cdots\!54\)\( p^{2} T^{2} - 133542942408 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 - 10491654264748 T + \)\(79\!\cdots\!82\)\( p T^{2} - 10491654264748 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 + 12405177148284 p T + \)\(17\!\cdots\!98\)\( p^{2} T^{2} + 12405177148284 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 + 907382448537944 T + \)\(32\!\cdots\!42\)\( p T^{2} + 907382448537944 p^{23} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 - 10116923323892112 T + \)\(67\!\cdots\!94\)\( T^{2} - 10116923323892112 p^{23} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 - 637198398109548 p T + \)\(81\!\cdots\!38\)\( T^{2} - 637198398109548 p^{24} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 + 272793622592745488 T + \)\(55\!\cdots\!82\)\( T^{2} + 272793622592745488 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 + 478995036787364 T + \)\(91\!\cdots\!26\)\( T^{2} + 478995036787364 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 - 5555714961308771412 T + \)\(17\!\cdots\!22\)\( T^{2} - 5555714961308771412 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 + 1198322147609320040 T - \)\(25\!\cdots\!10\)\( T^{2} + 1198322147609320040 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 - 26308565672855777280 T + \)\(74\!\cdots\!90\)\( T^{2} - 26308565672855777280 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 + 41127501899415224628 T + \)\(64\!\cdots\!74\)\( T^{2} + 41127501899415224628 p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 + \)\(30\!\cdots\!96\)\( T + \)\(11\!\cdots\!58\)\( T^{2} + \)\(30\!\cdots\!96\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 - \)\(59\!\cdots\!20\)\( T + \)\(26\!\cdots\!38\)\( T^{2} - \)\(59\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 + \)\(16\!\cdots\!88\)\( T + \)\(24\!\cdots\!62\)\( T^{2} + \)\(16\!\cdots\!88\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 + \)\(38\!\cdots\!76\)\( T + \)\(28\!\cdots\!66\)\( T^{2} + \)\(38\!\cdots\!76\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 + \)\(28\!\cdots\!28\)\( T + \)\(13\!\cdots\!54\)\( T^{2} + \)\(28\!\cdots\!28\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 - 21069388575313284880 T + \)\(88\!\cdots\!78\)\( T^{2} - 21069388575313284880 p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 + \)\(14\!\cdots\!24\)\( T + \)\(32\!\cdots\!42\)\( T^{2} + \)\(14\!\cdots\!24\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 + \)\(43\!\cdots\!64\)\( T + \)\(15\!\cdots\!18\)\( T^{2} + \)\(43\!\cdots\!64\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 - \)\(10\!\cdots\!92\)\( T + \)\(10\!\cdots\!62\)\( T^{2} - \)\(10\!\cdots\!92\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−21.02346685635690036292227301379, −19.62731892573877575492218507638, −18.90479510777121456839538943446, −18.16776800142464036292479820873, −17.31607371478141726122240935026, −16.92077067997102888633418088038, −15.72907926773758392455312018680, −14.62677916772822452245415391676, −13.31147812131123570052965132983, −12.73113147297691245499278761342, −11.24589917709964709030554567062, −10.80499844067584320318427704000, −8.966255851812802516898269592285, −8.901077629976462094078466883187, −6.95262491792968601961422457546, −5.96245679305929442058441056383, −4.44630644229121449340527660913, −3.97606202658624578829462454138, −1.38131907346156944497222112369, −0.46052755934266658061753194988, 0.46052755934266658061753194988, 1.38131907346156944497222112369, 3.97606202658624578829462454138, 4.44630644229121449340527660913, 5.96245679305929442058441056383, 6.95262491792968601961422457546, 8.901077629976462094078466883187, 8.966255851812802516898269592285, 10.80499844067584320318427704000, 11.24589917709964709030554567062, 12.73113147297691245499278761342, 13.31147812131123570052965132983, 14.62677916772822452245415391676, 15.72907926773758392455312018680, 16.92077067997102888633418088038, 17.31607371478141726122240935026, 18.16776800142464036292479820873, 18.90479510777121456839538943446, 19.62731892573877575492218507638, 21.02346685635690036292227301379

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