Properties

Degree $4$
Conductor $21609$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·3-s + 8·4-s + 18·5-s + 9·6-s + 45·8-s + 54·10-s + 36·11-s + 24·12-s − 68·13-s + 54·15-s + 135·16-s − 42·17-s + 124·19-s + 144·20-s + 108·22-s + 135·24-s + 125·25-s − 204·26-s − 27·27-s + 204·29-s + 162·30-s + 160·31-s + 360·32-s + 108·33-s − 126·34-s − 398·37-s + ⋯
L(s)  = 1  + 1.06·2-s + 0.577·3-s + 4-s + 1.60·5-s + 0.612·6-s + 1.98·8-s + 1.70·10-s + 0.986·11-s + 0.577·12-s − 1.45·13-s + 0.929·15-s + 2.10·16-s − 0.599·17-s + 1.49·19-s + 1.60·20-s + 1.04·22-s + 1.14·24-s + 25-s − 1.53·26-s − 0.192·27-s + 1.30·29-s + 0.985·30-s + 0.926·31-s + 1.98·32-s + 0.569·33-s − 0.635·34-s − 1.76·37-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: $1$
Motivic weight: \(3\)
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

\(L(2)\) \(\approx\) \(8.50441\)
\(L(\frac12)\) \(\approx\) \(8.50441\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 18 T + 199 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 36 T - 35 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 34 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 42 T - 3149 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 124 T + 8517 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 102 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 160 T - 4191 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 398 T + 107751 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 318 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 268 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 240 T - 46223 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 498 T + 99127 T^{2} - 498 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 132 T - 187955 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 398 T - 68577 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 92 T - 292299 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 720 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 502 T - 137013 T^{2} - 502 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 1024 T + 555537 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 204 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 354 T - 579653 T^{2} + 354 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 286 T + p^{3} T^{2} )^{2} \)
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

−13.30624328202591267409369612700, −12.29284372985196174583039302105, −11.84856313581763872790078226621, −11.74873248680591459369783079130, −10.64483611414046764699936459774, −10.25129650512381807787937548984, −9.830715025281998420737590511703, −9.537668631563427251780136069498, −8.570232827982493879575114273638, −8.196729264748521278575518083002, −7.07073344101855824087974020862, −7.03892974771903186343672133942, −6.37777405805474349100844875411, −5.51295160454589611163370816242, −4.98385409893098249065032521892, −4.62594930458511973997169298755, −3.51970122368449259145734785705, −2.85178390003630857089305525650, −1.93437003949380149975697212137, −1.46416920708817542687523475314, 1.46416920708817542687523475314, 1.93437003949380149975697212137, 2.85178390003630857089305525650, 3.51970122368449259145734785705, 4.62594930458511973997169298755, 4.98385409893098249065032521892, 5.51295160454589611163370816242, 6.37777405805474349100844875411, 7.03892974771903186343672133942, 7.07073344101855824087974020862, 8.196729264748521278575518083002, 8.570232827982493879575114273638, 9.537668631563427251780136069498, 9.830715025281998420737590511703, 10.25129650512381807787937548984, 10.64483611414046764699936459774, 11.74873248680591459369783079130, 11.84856313581763872790078226621, 12.29284372985196174583039302105, 13.30624328202591267409369612700

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