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Properties

Label 24-3380e12-1.1-c0e12-0-2
Degree $24$
Conductor $2.223\times 10^{42}$
Sign $1$
Analytic cond. $530.744$
Root an. cond. $1.29878$
Motivic weight $0$
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  − 2·3-s + 3·4-s + 3·9-s − 6·12-s + 3·16-s + 2·23-s − 6·25-s − 2·27-s + 2·29-s + 9·36-s + 2·43-s − 6·48-s − 49-s + 2·61-s − 2·64-s − 4·69-s + 12·75-s + 81-s − 4·87-s + 6·92-s − 18·100-s − 2·101-s − 4·103-s + 12·107-s − 6·108-s + 6·116-s + 6·121-s + ⋯
L(s)  = 1  − 2·3-s + 3·4-s + 3·9-s − 6·12-s + 3·16-s + 2·23-s − 6·25-s − 2·27-s + 2·29-s + 9·36-s + 2·43-s − 6·48-s − 49-s + 2·61-s − 2·64-s − 4·69-s + 12·75-s + 81-s − 4·87-s + 6·92-s − 18·100-s − 2·101-s − 4·103-s + 12·107-s − 6·108-s + 6·116-s + 6·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 5^{12} \cdot 13^{24}\)
Sign: $1$
Analytic conductor: \(530.744\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 5^{12} \cdot 13^{24} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.652081611\)
\(L(\frac12)\) \(\approx\) \(3.652081611\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{3} \)
5 \( ( 1 + T^{2} )^{6} \)
13 \( 1 \)
good3 \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \)
7 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
11 \( ( 1 - T^{2} + T^{4} )^{6} \)
17 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
19 \( ( 1 - T^{2} + T^{4} )^{6} \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
29 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
31 \( ( 1 + T^{2} )^{12} \)
37 \( ( 1 - T^{2} + T^{4} )^{6} \)
41 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
43 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
53 \( ( 1 - T )^{12}( 1 + T )^{12} \)
59 \( ( 1 - T^{2} + T^{4} )^{6} \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
67 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
71 \( ( 1 - T^{2} + T^{4} )^{6} \)
73 \( ( 1 + T^{2} )^{12} \)
79 \( ( 1 - T )^{12}( 1 + T )^{12} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
89 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
97 \( ( 1 - T^{2} + T^{4} )^{6} \)
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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

−2.89939325284227409844063259616, −2.77247460420880530447858801874, −2.51820558738696211067371205816, −2.48308321934877943424920698010, −2.37819317041807503626218672985, −2.35702418820719239043730764157, −2.33961915151079165590267788065, −2.26420305282857945330390063955, −2.08843404143042881270283718406, −2.01244266418160247925440200171, −1.98966659284296221469051126806, −1.85306377002315342007867076375, −1.77598558590967914869263832689, −1.76760095048315719694056695654, −1.70653180766348117051954374237, −1.56835154410693863282321618764, −1.51415988002894388174364802078, −1.21660043839012596588136241528, −1.19342960086011963789553649313, −1.02753243536270224154717360985, −0.945881162702571285306121609483, −0.817166871027631114444307414590, −0.70372537327768501312331637761, −0.57016178644690592563445887186, −0.41029900060435761315937134187, 0.41029900060435761315937134187, 0.57016178644690592563445887186, 0.70372537327768501312331637761, 0.817166871027631114444307414590, 0.945881162702571285306121609483, 1.02753243536270224154717360985, 1.19342960086011963789553649313, 1.21660043839012596588136241528, 1.51415988002894388174364802078, 1.56835154410693863282321618764, 1.70653180766348117051954374237, 1.76760095048315719694056695654, 1.77598558590967914869263832689, 1.85306377002315342007867076375, 1.98966659284296221469051126806, 2.01244266418160247925440200171, 2.08843404143042881270283718406, 2.26420305282857945330390063955, 2.33961915151079165590267788065, 2.35702418820719239043730764157, 2.37819317041807503626218672985, 2.48308321934877943424920698010, 2.51820558738696211067371205816, 2.77247460420880530447858801874, 2.89939325284227409844063259616

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

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