Properties

Label 4-1216e2-1.1-c1e2-0-9
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
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  − 3-s − 5-s + 3·9-s + 8·11-s − 13-s + 15-s − 3·17-s + 8·19-s − 5·23-s + 5·25-s − 8·27-s + 7·29-s + 8·31-s − 8·33-s − 20·37-s + 39-s + 5·41-s − 5·43-s − 3·45-s + 7·47-s − 14·49-s + 3·51-s + 11·53-s − 8·55-s − 8·57-s + 3·59-s + 11·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 9-s + 2.41·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 1.83·19-s − 1.04·23-s + 25-s − 1.53·27-s + 1.29·29-s + 1.43·31-s − 1.39·33-s − 3.28·37-s + 0.160·39-s + 0.780·41-s − 0.762·43-s − 0.447·45-s + 1.02·47-s − 2·49-s + 0.420·51-s + 1.51·53-s − 1.07·55-s − 1.05·57-s + 0.390·59-s + 1.40·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.038716461\)
\(L(\frac12)\) \(\approx\) \(2.038716461\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 11 T + 50 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−10.12974116851784177510739424791, −9.535410791987473980052116502366, −9.149145462644269915291853308443, −8.730656008192374881332195779733, −8.441077738281850344115536236223, −7.82648993991234874058193208123, −7.32789131671831894718956048886, −6.87775854573007508450641997933, −6.73413549737044777708227204023, −6.39919526618954127540283321712, −5.69132326049427070035321836526, −5.30244719325971240695223039780, −4.72791856684776711902182837348, −4.26937653309723728699824070656, −3.96214577637191392787099964113, −3.42643847339027405608035657755, −2.89103846145515756479984791114, −1.83657513146358670977229837723, −1.41569620587877553977564110850, −0.70474159750067721955262695646, 0.70474159750067721955262695646, 1.41569620587877553977564110850, 1.83657513146358670977229837723, 2.89103846145515756479984791114, 3.42643847339027405608035657755, 3.96214577637191392787099964113, 4.26937653309723728699824070656, 4.72791856684776711902182837348, 5.30244719325971240695223039780, 5.69132326049427070035321836526, 6.39919526618954127540283321712, 6.73413549737044777708227204023, 6.87775854573007508450641997933, 7.32789131671831894718956048886, 7.82648993991234874058193208123, 8.441077738281850344115536236223, 8.730656008192374881332195779733, 9.149145462644269915291853308443, 9.535410791987473980052116502366, 10.12974116851784177510739424791

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