Properties

Label 4-147e2-1.1-c3e2-0-4
Degree $4$
Conductor $21609$
Sign $1$
Analytic cond. $75.2257$
Root an. cond. $2.94504$
Motivic weight $3$
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  + 9·3-s − 8·4-s + 54·9-s − 72·12-s − 270·19-s + 125·25-s + 243·27-s + 270·31-s − 432·36-s + 110·37-s + 1.04e3·43-s − 2.43e3·57-s + 1.62e3·61-s + 512·64-s + 880·67-s + 648·73-s + 1.12e3·75-s + 2.16e3·76-s − 884·79-s + 729·81-s + 2.43e3·93-s − 1.00e3·100-s − 1.78e3·103-s − 1.94e3·108-s − 646·109-s + 990·111-s − 1.33e3·121-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s + 2·9-s − 1.73·12-s − 3.26·19-s + 25-s + 1.73·27-s + 1.56·31-s − 2·36-s + 0.488·37-s + 3.68·43-s − 5.64·57-s + 3.40·61-s + 64-s + 1.60·67-s + 1.03·73-s + 1.73·75-s + 3.26·76-s − 1.25·79-s + 81-s + 2.70·93-s − 100-s − 1.70·103-s − 1.73·108-s − 0.567·109-s + 0.846·111-s − 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21609\)    =    \(3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(75.2257\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{147} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21609,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.976249198\)
\(L(\frac12)\) \(\approx\) \(2.976249198\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - p^{2} T + p^{3} T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + p^{3} T^{2} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + p^{3} T^{2} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 70 T + p^{3} T^{2} )( 1 + 70 T + p^{3} T^{2} ) \)
17$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 107 T + p^{3} T^{2} )( 1 + 163 T + p^{3} T^{2} ) \)
23$C_2^2$ \( 1 + p^{3} T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 289 T + p^{3} T^{2} )( 1 + 19 T + p^{3} T^{2} ) \)
37$C_2$ \( ( 1 - 433 T + p^{3} T^{2} )( 1 + 323 T + p^{3} T^{2} ) \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 520 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + p^{3} T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 901 T + p^{3} T^{2} )( 1 - 719 T + p^{3} T^{2} ) \)
67$C_2$ \( ( 1 - 1007 T + p^{3} T^{2} )( 1 + 127 T + p^{3} T^{2} ) \)
71$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 919 T + p^{3} T^{2} )( 1 + 271 T + p^{3} T^{2} ) \)
79$C_2$ \( ( 1 - 503 T + p^{3} T^{2} )( 1 + 1387 T + p^{3} T^{2} ) \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 1330 T + p^{3} T^{2} )( 1 + 1330 T + p^{3} T^{2} ) \)
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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

−12.97237788051604285921999944468, −12.76429235265236511575299091310, −12.11467778887022233795233403886, −11.07084031480458158138030545482, −10.76276685109471222525457516020, −10.09833285523190914002744566774, −9.605011104500332028826719638495, −9.055513163319753054120133996004, −8.700510170234253245939204825906, −8.231482053800568090433631156479, −7.998988012514976192620873524052, −6.90874503228987059552471789283, −6.65184892002337987773017839813, −5.68430571937809836570203800509, −4.61938350253875917193651749385, −4.21038479497882429479439891498, −3.85000308351963722934647797393, −2.52046545616122236116475578433, −2.34254653060397117794089619841, −0.78743987367623522268325718696, 0.78743987367623522268325718696, 2.34254653060397117794089619841, 2.52046545616122236116475578433, 3.85000308351963722934647797393, 4.21038479497882429479439891498, 4.61938350253875917193651749385, 5.68430571937809836570203800509, 6.65184892002337987773017839813, 6.90874503228987059552471789283, 7.998988012514976192620873524052, 8.231482053800568090433631156479, 8.700510170234253245939204825906, 9.055513163319753054120133996004, 9.605011104500332028826719638495, 10.09833285523190914002744566774, 10.76276685109471222525457516020, 11.07084031480458158138030545482, 12.11467778887022233795233403886, 12.76429235265236511575299091310, 12.97237788051604285921999944468

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