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Properties

Label 24-3724e12-1.1-c0e12-0-2
Degree $24$
Conductor $7.114\times 10^{42}$
Sign $1$
Analytic cond. $1698.24$
Root an. cond. $1.36327$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 7-s + 9-s + 13·11-s + 13·17-s − 6·19-s − 23-s + 3·25-s + 2·35-s − 4·43-s + 2·45-s − 47-s + 49-s + 26·55-s − 61-s + 63-s − 73-s + 13·77-s + 81-s + 2·83-s + 26·85-s − 12·95-s + 13·99-s − 8·101-s − 2·115-s + 13·119-s + 91·121-s + ⋯
L(s)  = 1  + 2·5-s + 7-s + 9-s + 13·11-s + 13·17-s − 6·19-s − 23-s + 3·25-s + 2·35-s − 4·43-s + 2·45-s − 47-s + 49-s + 26·55-s − 61-s + 63-s − 73-s + 13·77-s + 81-s + 2·83-s + 26·85-s − 12·95-s + 13·99-s − 8·101-s − 2·115-s + 13·119-s + 91·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 7^{24} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(1698.24\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 7^{24} \cdot 19^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(84.22212872\)
\(L(\frac12)\) \(\approx\) \(84.22212872\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

Imaginary part of the first few zeros on the critical line

−2.93689531258565180293756332406, −2.75754281777982667396991546772, −2.47637788272686510947371818493, −2.36707569047293314107486362276, −2.36504792047212276565044186775, −2.34251241415137264576270793660, −2.23858672416110247516125543086, −2.14037499429443075240159178424, −1.85371954292119635191926526194, −1.82426946175875713491119539182, −1.80916989962132957335363075105, −1.73174979884162949069378551601, −1.70807957493952314038705740703, −1.58483494639346233134762855115, −1.43317530542905081678152717267, −1.35142123893313140028802424287, −1.29934061426430815747899381288, −1.28018722407742017471674057945, −1.19281341270905322285417476442, −1.17181630180296655943325297226, −1.13607967126861699214337220069, −1.03967764171324950056753456484, −1.02985839075414102929696797308, −0.925553352201926599715407459906, −0.72245616435507809427634483158, 0.72245616435507809427634483158, 0.925553352201926599715407459906, 1.02985839075414102929696797308, 1.03967764171324950056753456484, 1.13607967126861699214337220069, 1.17181630180296655943325297226, 1.19281341270905322285417476442, 1.28018722407742017471674057945, 1.29934061426430815747899381288, 1.35142123893313140028802424287, 1.43317530542905081678152717267, 1.58483494639346233134762855115, 1.70807957493952314038705740703, 1.73174979884162949069378551601, 1.80916989962132957335363075105, 1.82426946175875713491119539182, 1.85371954292119635191926526194, 2.14037499429443075240159178424, 2.23858672416110247516125543086, 2.34251241415137264576270793660, 2.36504792047212276565044186775, 2.36707569047293314107486362276, 2.47637788272686510947371818493, 2.75754281777982667396991546772, 2.93689531258565180293756332406

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

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