↑

Properties

Label 24-3332e12-1.1-c0e12-0-2
Degree $24$
Conductor $1.873\times 10^{42}$
Sign $1$
Analytic cond. $447.038$
Root an. cond. $1.28952$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 13·3-s + 4-s + 13·6-s − 2·7-s + 91·9-s − 8·11-s + 13·12-s − 5·13-s − 2·14-s + 17-s + 91·18-s − 26·21-s − 8·22-s + 2·23-s + 25-s − 5·26-s + 454·27-s − 2·28-s + 2·31-s − 104·33-s + 34-s + 91·36-s − 65·39-s − 26·42-s − 8·44-s + 2·46-s + ⋯
L(s)  = 1  + 2-s + 13·3-s + 4-s + 13·6-s − 2·7-s + 91·9-s − 8·11-s + 13·12-s − 5·13-s − 2·14-s + 17-s + 91·18-s − 26·21-s − 8·22-s + 2·23-s + 25-s − 5·26-s + 454·27-s − 2·28-s + 2·31-s − 104·33-s + 34-s + 91·36-s − 65·39-s − 26·42-s − 8·44-s + 2·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 7^{24} \cdot 17^{12}\)
Sign: $1$
Analytic conductor: \(447.038\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 7^{24} \cdot 17^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(620.4272940\)
\(L(\frac12)\) \(\approx\) \(620.4272940\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
17 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
good3 \( ( 1 - T )^{12}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
5 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
11 \( ( 1 + T + T^{2} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
13 \( ( 1 + T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
19 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
31 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
67 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
97 \( ( 1 - T )^{12}( 1 + T )^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.85115601639572777960728600911, −2.71919204563519525814081229379, −2.68801468291710407811873272939, −2.58272558800112628394128685031, −2.49656614761230653677973037554, −2.45987664307698134294671040315, −2.41003159538276845698564349626, −2.40181367480797648503921466942, −2.37980056555390516334230007232, −2.24987563009292359872437994132, −2.23748452126035914842930989058, −2.22545697546608737360943963645, −2.20176569876881827714066831947, −2.16847270106598409535094042219, −1.88249436121175035318284909304, −1.80198474382053369787654802187, −1.54217661353830773718073996063, −1.54172678863558045964134210776, −1.35612012906003454602534806278, −1.27134875628648076399096018637, −1.20486968366875468888792793000, −1.08062908940377190063004844794, −1.07231411769789858903287804612, −0.956515605941062994820044242250, −0.38138126468301668637445155566, 0.38138126468301668637445155566, 0.956515605941062994820044242250, 1.07231411769789858903287804612, 1.08062908940377190063004844794, 1.20486968366875468888792793000, 1.27134875628648076399096018637, 1.35612012906003454602534806278, 1.54172678863558045964134210776, 1.54217661353830773718073996063, 1.80198474382053369787654802187, 1.88249436121175035318284909304, 2.16847270106598409535094042219, 2.20176569876881827714066831947, 2.22545697546608737360943963645, 2.23748452126035914842930989058, 2.24987563009292359872437994132, 2.37980056555390516334230007232, 2.40181367480797648503921466942, 2.41003159538276845698564349626, 2.45987664307698134294671040315, 2.49656614761230653677973037554, 2.58272558800112628394128685031, 2.68801468291710407811873272939, 2.71919204563519525814081229379, 2.85115601639572777960728600911

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

Plot not available for L-functions of degree greater than 10.