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Properties

Label 4-3e2-1.1-c21e2-0-0
Degree $4$
Conductor $9$
Sign $1$
Analytic cond. $70.2968$
Root an. cond. $2.89556$
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  + 666·2-s + 1.18e5·3-s − 1.28e6·4-s + 9.96e5·5-s + 7.86e7·6-s + 6.79e8·7-s − 6.11e8·8-s + 1.04e10·9-s + 6.63e8·10-s + 2.19e11·11-s − 1.51e11·12-s − 4.84e10·13-s + 4.52e11·14-s + 1.17e11·15-s − 1.65e12·16-s − 1.13e13·17-s + 6.96e12·18-s + 1.19e13·19-s − 1.28e12·20-s + 8.02e13·21-s + 1.46e14·22-s − 1.46e14·23-s − 7.21e13·24-s − 4.78e14·25-s − 3.22e13·26-s + 8.23e14·27-s − 8.74e14·28-s + ⋯
L(s)  = 1  + 0.459·2-s + 1.15·3-s − 0.613·4-s + 0.0456·5-s + 0.531·6-s + 0.909·7-s − 0.201·8-s + 9-s + 0.0209·10-s + 2.55·11-s − 0.707·12-s − 0.0975·13-s + 0.418·14-s + 0.0527·15-s − 0.375·16-s − 1.36·17-s + 0.459·18-s + 0.447·19-s − 0.0279·20-s + 1.05·21-s + 1.17·22-s − 0.737·23-s − 0.232·24-s − 1.00·25-s − 0.0448·26-s + 0.769·27-s − 0.557·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(70.2968\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(4.778385969\)
\(L(\frac12)\) \(\approx\) \(4.778385969\)
\(L(\frac{23}{2})\) 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)$
bad3$C_1$ \( ( 1 - p^{10} T )^{2} \)
good2$D_{4}$ \( 1 - 333 p T + 54041 p^{5} T^{2} - 333 p^{22} T^{3} + p^{42} T^{4} \)
5$D_{4}$ \( 1 - 996876 T + 19189088648254 p^{2} T^{2} - 996876 p^{21} T^{3} + p^{42} T^{4} \)
7$D_{4}$ \( 1 - 97128016 p T + 23414651512456686 p^{2} T^{2} - 97128016 p^{22} T^{3} + p^{42} T^{4} \)
11$D_{4}$ \( 1 - 19988102088 p T + \)\(20\!\cdots\!54\)\( p^{2} T^{2} - 19988102088 p^{22} T^{3} + p^{42} T^{4} \)
13$D_{4}$ \( 1 + 48468909956 T - \)\(66\!\cdots\!82\)\( p T^{2} + 48468909956 p^{21} T^{3} + p^{42} T^{4} \)
17$D_{4}$ \( 1 + 666678178908 p T + \)\(56\!\cdots\!22\)\( p^{2} T^{2} + 666678178908 p^{22} T^{3} + p^{42} T^{4} \)
19$D_{4}$ \( 1 - 629504474296 p T + \)\(48\!\cdots\!38\)\( p^{2} T^{2} - 629504474296 p^{22} T^{3} + p^{42} T^{4} \)
23$D_{4}$ \( 1 + 146508390063504 T + \)\(70\!\cdots\!46\)\( T^{2} + 146508390063504 p^{21} T^{3} + p^{42} T^{4} \)
29$D_{4}$ \( 1 + 1798520043674052 T + \)\(75\!\cdots\!58\)\( T^{2} + 1798520043674052 p^{21} T^{3} + p^{42} T^{4} \)
31$D_{4}$ \( 1 - 11169107526944992 T + \)\(65\!\cdots\!62\)\( T^{2} - 11169107526944992 p^{21} T^{3} + p^{42} T^{4} \)
37$D_{4}$ \( 1 - 12736264858660012 T + \)\(11\!\cdots\!54\)\( T^{2} - 12736264858660012 p^{21} T^{3} + p^{42} T^{4} \)
41$D_{4}$ \( 1 - 122972020616468052 T + \)\(11\!\cdots\!02\)\( T^{2} - 122972020616468052 p^{21} T^{3} + p^{42} T^{4} \)
43$D_{4}$ \( 1 - 288455418162270040 T + \)\(60\!\cdots\!30\)\( T^{2} - 288455418162270040 p^{21} T^{3} + p^{42} T^{4} \)
47$D_{4}$ \( 1 - 837243745741596960 T + \)\(43\!\cdots\!10\)\( T^{2} - 837243745741596960 p^{21} T^{3} + p^{42} T^{4} \)
53$D_{4}$ \( 1 + 43007964012775764 T + \)\(30\!\cdots\!46\)\( T^{2} + 43007964012775764 p^{21} T^{3} + p^{42} T^{4} \)
59$D_{4}$ \( 1 + 3523823330903857224 T + \)\(22\!\cdots\!98\)\( T^{2} + 3523823330903857224 p^{21} T^{3} + p^{42} T^{4} \)
61$D_{4}$ \( 1 + 1779023128451013860 T + \)\(54\!\cdots\!38\)\( T^{2} + 1779023128451013860 p^{21} T^{3} + p^{42} T^{4} \)
67$D_{4}$ \( 1 + 16454068667621610296 T + \)\(48\!\cdots\!38\)\( T^{2} + 16454068667621610296 p^{21} T^{3} + p^{42} T^{4} \)
71$D_{4}$ \( 1 - 17379227131150420944 T + \)\(15\!\cdots\!26\)\( T^{2} - 17379227131150420944 p^{21} T^{3} + p^{42} T^{4} \)
73$D_{4}$ \( 1 - 50891146268473989076 T + \)\(15\!\cdots\!06\)\( T^{2} - 50891146268473989076 p^{21} T^{3} + p^{42} T^{4} \)
79$D_{4}$ \( 1 + 54055785594190591040 T + \)\(14\!\cdots\!58\)\( T^{2} + 54055785594190591040 p^{21} T^{3} + p^{42} T^{4} \)
83$D_{4}$ \( 1 - \)\(11\!\cdots\!88\)\( T + \)\(39\!\cdots\!78\)\( T^{2} - \)\(11\!\cdots\!88\)\( p^{21} T^{3} + p^{42} T^{4} \)
89$D_{4}$ \( 1 - \)\(22\!\cdots\!24\)\( T + \)\(17\!\cdots\!98\)\( T^{2} - \)\(22\!\cdots\!24\)\( p^{21} T^{3} + p^{42} T^{4} \)
97$D_{4}$ \( 1 + \)\(12\!\cdots\!36\)\( T + \)\(78\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!36\)\( p^{21} T^{3} + p^{42} T^{4} \)
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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

−21.49544547100370862712309988850, −20.48939555480983151977093474340, −19.75672159148241859643238816434, −19.15820348472221603750848389109, −17.80162857818988804969123027532, −17.32901364553384648815509377694, −15.78466180859828723067671639827, −14.78307903655031106595731429575, −13.89971987886233398034241738487, −13.83399071979712557391409054935, −12.26122727322474893680547678385, −11.24907079689894967943496641418, −9.415981201009002988706469216919, −8.985468658839268506463878719954, −7.74641459196408745592655202908, −6.34537009998925288438717855760, −4.31001648806194752940832954182, −4.09558009622312387136400257069, −2.25684997475839112161469020857, −1.12394556401439308955174821673, 1.12394556401439308955174821673, 2.25684997475839112161469020857, 4.09558009622312387136400257069, 4.31001648806194752940832954182, 6.34537009998925288438717855760, 7.74641459196408745592655202908, 8.985468658839268506463878719954, 9.415981201009002988706469216919, 11.24907079689894967943496641418, 12.26122727322474893680547678385, 13.83399071979712557391409054935, 13.89971987886233398034241738487, 14.78307903655031106595731429575, 15.78466180859828723067671639827, 17.32901364553384648815509377694, 17.80162857818988804969123027532, 19.15820348472221603750848389109, 19.75672159148241859643238816434, 20.48939555480983151977093474340, 21.49544547100370862712309988850

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