Properties

Degree $4$
Conductor $21609$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 4-s − 4·5-s + 4·6-s + 3·9-s + 8·10-s − 4·11-s − 2·12-s − 8·13-s + 8·15-s + 16-s − 4·17-s − 6·18-s − 4·20-s + 8·22-s − 4·23-s + 4·25-s + 16·26-s − 4·27-s − 8·29-s − 16·30-s + 8·31-s + 2·32-s + 8·33-s + 8·34-s + 3·36-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 1.63·6-s + 9-s + 2.52·10-s − 1.20·11-s − 0.577·12-s − 2.21·13-s + 2.06·15-s + 1/4·16-s − 0.970·17-s − 1.41·18-s − 0.894·20-s + 1.70·22-s − 0.834·23-s + 4/5·25-s + 3.13·26-s − 0.769·27-s − 1.48·29-s − 2.92·30-s + 1.43·31-s + 0.353·32-s + 1.39·33-s + 1.37·34-s + 1/2·36-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 64 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 208 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.24862634564251127101582445586, −12.11072490375510402254582724345, −11.95663633100051336715682572906, −11.22502223928612795455263620771, −10.62915661160969898296205034709, −10.38466435605565616574837825783, −9.535229732165943745817356379510, −9.527103915730750902096128604275, −8.426948330020650326983175483803, −8.134790574271130183505669457339, −7.42349770612555731170106759369, −7.41122021895125240898746911159, −6.52613855611675000400346196711, −5.65388414877058007429182791770, −4.78470749537121392341897557520, −4.57935404398144771969035979627, −3.53087914044105296455979885058, −2.29411554390972943087601573623, 0, 0, 2.29411554390972943087601573623, 3.53087914044105296455979885058, 4.57935404398144771969035979627, 4.78470749537121392341897557520, 5.65388414877058007429182791770, 6.52613855611675000400346196711, 7.41122021895125240898746911159, 7.42349770612555731170106759369, 8.134790574271130183505669457339, 8.426948330020650326983175483803, 9.527103915730750902096128604275, 9.535229732165943745817356379510, 10.38466435605565616574837825783, 10.62915661160969898296205034709, 11.22502223928612795455263620771, 11.95663633100051336715682572906, 12.11072490375510402254582724345, 12.24862634564251127101582445586

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