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Properties

Label 4-3e4-1.1-c21e2-0-0
Degree $4$
Conductor $81$
Sign $1$
Analytic cond. $632.671$
Root an. cond. $5.01527$
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 − 1.28e6·4-s − 9.96e5·5-s + 6.79e8·7-s + 6.11e8·8-s + 6.63e8·10-s − 2.19e11·11-s − 4.84e10·13-s − 4.52e11·14-s − 1.65e12·16-s + 1.13e13·17-s + 1.19e13·19-s + 1.28e12·20-s + 1.46e14·22-s + 1.46e14·23-s − 4.78e14·25-s + 3.22e13·26-s − 8.74e14·28-s + 1.79e15·29-s + 1.11e16·31-s + 4.76e15·32-s − 7.54e15·34-s − 6.77e14·35-s + 1.27e16·37-s − 7.96e15·38-s − 6.09e14·40-s − 1.22e17·41-s + ⋯
L(s)  = 1  − 0.459·2-s − 0.613·4-s − 0.0456·5-s + 0.909·7-s + 0.201·8-s + 0.0209·10-s − 2.55·11-s − 0.0975·13-s − 0.418·14-s − 0.375·16-s + 1.36·17-s + 0.447·19-s + 0.0279·20-s + 1.17·22-s + 0.737·23-s − 1.00·25-s + 0.0448·26-s − 0.557·28-s + 0.793·29-s + 2.44·31-s + 0.748·32-s − 0.627·34-s − 0.0415·35-s + 0.435·37-s − 0.205·38-s − 0.00918·40-s − 1.43·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $1$
Analytic conductor: \(632.671\)
Root analytic conductor: \(5.01527\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 81,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(0.9685226229\)
\(L(\frac12)\) \(\approx\) \(0.9685226229\)
\(L(\frac{23}{2})\) 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 \( 1 \)
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

−16.28840568283866407544562494979, −15.71950741249817679745562138072, −15.05653681714102024596849310279, −14.07258330141691945639913971775, −13.52145812722002435240906711502, −12.81626208261011362283744084379, −11.85078522177591995827296343640, −11.09432143062809884988752872425, −10.01631058338208857695815856530, −9.881748268042164303783479361149, −8.327728500524252218697218219586, −8.174762702916693838669892475364, −7.40498662748837978922671546410, −5.98136708677993779512077036755, −4.93956968454570441006330021795, −4.75658686997557929485351830652, −3.14767765435172049491292905417, −2.47909800498750747947282194614, −1.24415797490543756439724998456, −0.39441388374342991176490321881, 0.39441388374342991176490321881, 1.24415797490543756439724998456, 2.47909800498750747947282194614, 3.14767765435172049491292905417, 4.75658686997557929485351830652, 4.93956968454570441006330021795, 5.98136708677993779512077036755, 7.40498662748837978922671546410, 8.174762702916693838669892475364, 8.327728500524252218697218219586, 9.881748268042164303783479361149, 10.01631058338208857695815856530, 11.09432143062809884988752872425, 11.85078522177591995827296343640, 12.81626208261011362283744084379, 13.52145812722002435240906711502, 14.07258330141691945639913971775, 15.05653681714102024596849310279, 15.71950741249817679745562138072, 16.28840568283866407544562494979

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