Properties

Label 4-3e6-1.1-c2e2-0-0
Degree $4$
Conductor $729$
Sign $1$
Analytic cond. $0.541249$
Root an. cond. $0.857727$
Motivic weight $2$
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  − 4-s + 10·7-s − 20·13-s − 15·16-s − 32·19-s + 41·25-s − 10·28-s − 2·31-s + 40·37-s + 100·43-s − 23·49-s + 20·52-s − 152·61-s + 31·64-s − 20·67-s + 130·73-s + 32·76-s + 28·79-s − 200·91-s − 170·97-s − 41·100-s + 340·103-s + 328·109-s − 150·112-s + 17·121-s + 2·124-s + 127-s + ⋯
L(s)  = 1  − 1/4·4-s + 10/7·7-s − 1.53·13-s − 0.937·16-s − 1.68·19-s + 1.63·25-s − 0.357·28-s − 0.0645·31-s + 1.08·37-s + 2.32·43-s − 0.469·49-s + 5/13·52-s − 2.49·61-s + 0.484·64-s − 0.298·67-s + 1.78·73-s + 8/19·76-s + 0.354·79-s − 2.19·91-s − 1.75·97-s − 0.409·100-s + 3.30·103-s + 3.00·109-s − 1.33·112-s + 0.140·121-s + 1/62·124-s + 0.00787·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(0.541249\)
Root analytic conductor: \(0.857727\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 729,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8986965568\)
\(L(\frac12)\) \(\approx\) \(0.8986965568\)
\(L(2)\) 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$C_2^2$ \( 1 + T^{2} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 41 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 17 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 254 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 914 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 782 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 20 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 238 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 50 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4382 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4889 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6062 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 76 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 1982 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 65 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 13769 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 7742 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 85 T + p^{2} T^{2} )^{2} \)
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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

−17.36808523137500384684965942861, −16.95867641883435699347127869064, −16.44271949144973078182113659631, −15.32702670319768498329699627984, −15.02205089555784189285554512850, −14.21316958405189901201418238593, −14.14876383614042419244364506857, −12.78777173527289554999790494628, −12.65414210033438109237929814485, −11.65390726870685934002347607507, −10.99282694792126830280638291561, −10.52950361117322614987594461701, −9.420708282430980154915816688071, −8.829029835700150106568585551216, −7.981613995047342787015509059394, −7.28542109883854820063536672271, −6.24922237424642260720064150063, −4.83051333614660063586439682390, −4.53354833528689371647994175977, −2.36734171042264431836895026524, 2.36734171042264431836895026524, 4.53354833528689371647994175977, 4.83051333614660063586439682390, 6.24922237424642260720064150063, 7.28542109883854820063536672271, 7.981613995047342787015509059394, 8.829029835700150106568585551216, 9.420708282430980154915816688071, 10.52950361117322614987594461701, 10.99282694792126830280638291561, 11.65390726870685934002347607507, 12.65414210033438109237929814485, 12.78777173527289554999790494628, 14.14876383614042419244364506857, 14.21316958405189901201418238593, 15.02205089555784189285554512850, 15.32702670319768498329699627984, 16.44271949144973078182113659631, 16.95867641883435699347127869064, 17.36808523137500384684965942861

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