Properties

Label 4-54e2-1.1-c8e2-0-0
Degree $4$
Conductor $2916$
Sign $1$
Analytic cond. $483.931$
Root an. cond. $4.69024$
Motivic weight $8$
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  − 128·4-s − 4.13e3·7-s + 1.61e4·13-s + 1.63e4·16-s − 4.53e5·19-s + 3.20e5·25-s + 5.28e5·28-s + 1.65e6·31-s + 2.68e6·37-s − 1.22e7·43-s + 1.26e6·49-s − 2.06e6·52-s − 2.99e7·61-s − 2.09e6·64-s − 2.00e7·67-s − 4.65e7·73-s + 5.80e7·76-s + 2.85e7·79-s − 6.66e7·91-s − 8.11e7·97-s − 4.10e7·100-s − 2.74e8·103-s − 8.58e7·109-s − 6.76e7·112-s + 3.84e8·121-s − 2.11e8·124-s + 127-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.72·7-s + 0.564·13-s + 1/4·16-s − 3.47·19-s + 0.820·25-s + 0.860·28-s + 1.78·31-s + 1.43·37-s − 3.59·43-s + 0.219·49-s − 0.282·52-s − 2.16·61-s − 1/8·64-s − 0.994·67-s − 1.63·73-s + 1.73·76-s + 0.732·79-s − 0.971·91-s − 0.916·97-s − 0.410·100-s − 2.43·103-s − 0.608·109-s − 0.430·112-s + 1.79·121-s − 0.894·124-s + 5.98·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(483.931\)
Root analytic conductor: \(4.69024\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2916,\ (\ :4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.007037124938\)
\(L(\frac12)\) \(\approx\) \(0.007037124938\)
\(L(5)\) 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$C_2$ \( 1 + p^{7} T^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 12818 p^{2} T^{2} + p^{16} T^{4} \)
7$C_2$ \( ( 1 + 295 p T + p^{8} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 384462530 T^{2} + p^{16} T^{4} \)
13$C_2$ \( ( 1 - 8063 T + p^{8} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 13485535490 T^{2} + p^{16} T^{4} \)
19$C_2$ \( ( 1 + 226609 T + p^{8} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 20955649154 T^{2} + p^{16} T^{4} \)
29$C_2^2$ \( 1 - 145584434 p^{2} T^{2} + p^{16} T^{4} \)
31$C_2$ \( ( 1 - 826370 T + p^{8} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 1344575 T + p^{8} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 10986038337406 T^{2} + p^{16} T^{4} \)
43$C_2$ \( ( 1 + 6147742 T + p^{8} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 12685263482114 T^{2} + p^{16} T^{4} \)
53$C_2^2$ \( 1 - 123929317870274 T^{2} + p^{16} T^{4} \)
59$C_2^2$ \( 1 - 293436242931650 T^{2} + p^{16} T^{4} \)
61$C_2$ \( ( 1 + 14985697 T + p^{8} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 10023697 T + p^{8} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 777369936279166 T^{2} + p^{16} T^{4} \)
73$C_2$ \( ( 1 + 23261569 T + p^{8} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 14267183 T + p^{8} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 3194735948169794 T^{2} + p^{16} T^{4} \)
89$C_2^2$ \( 1 + 5372002023507070 T^{2} + p^{16} T^{4} \)
97$C_2$ \( ( 1 + 40571617 T + p^{8} T^{2} )^{2} \)
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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

−13.53452531560391858061370678342, −13.36009667554431964317882168602, −12.89539181101130305240661428405, −12.42472790942421970277713339427, −11.73559560542652893167290662129, −10.78695639459430025697913683898, −10.41505064699231845020114047792, −9.816006207693416329938664046144, −9.216199875780054088331923826848, −8.362324261417414539426944629797, −8.305236224768396701721861721573, −6.87077101950411592059378679206, −6.27708731810099713176079237022, −6.22114443536005335983964391630, −4.75056911455626054007110796660, −4.24688015296670408978462999963, −3.31480778809900898652660708313, −2.63939802955244067024854563713, −1.44144272606912720024728855532, −0.02970082126334810075167608930, 0.02970082126334810075167608930, 1.44144272606912720024728855532, 2.63939802955244067024854563713, 3.31480778809900898652660708313, 4.24688015296670408978462999963, 4.75056911455626054007110796660, 6.22114443536005335983964391630, 6.27708731810099713176079237022, 6.87077101950411592059378679206, 8.305236224768396701721861721573, 8.362324261417414539426944629797, 9.216199875780054088331923826848, 9.816006207693416329938664046144, 10.41505064699231845020114047792, 10.78695639459430025697913683898, 11.73559560542652893167290662129, 12.42472790942421970277713339427, 12.89539181101130305240661428405, 13.36009667554431964317882168602, 13.53452531560391858061370678342

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