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Properties

Label 24-3675e12-1.1-c0e12-0-2
Degree $24$
Conductor $6.069\times 10^{42}$
Sign $1$
Analytic cond. $1448.68$
Root an. cond. $1.35427$
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  − 3-s + 4-s − 7-s + 9-s − 12-s − 2·13-s + 16-s − 19-s + 21-s − 28-s + 2·31-s + 36-s + 8·37-s + 2·39-s + 4·43-s − 48-s + 49-s − 2·52-s + 57-s + 13·61-s − 63-s + 67-s + 73-s − 76-s − 79-s + 84-s + 2·91-s + ⋯
L(s)  = 1  − 3-s + 4-s − 7-s + 9-s − 12-s − 2·13-s + 16-s − 19-s + 21-s − 28-s + 2·31-s + 36-s + 8·37-s + 2·39-s + 4·43-s − 48-s + 49-s − 2·52-s + 57-s + 13·61-s − 63-s + 67-s + 73-s − 76-s − 79-s + 84-s + 2·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 5^{24} \cdot 7^{24}\)
Sign: $1$
Analytic conductor: \(1448.68\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 5^{24} \cdot 7^{24} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.682771500\)
\(L(\frac12)\) \(\approx\) \(5.682771500\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
5 \( 1 \)
7 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
11 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
13 \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \)
17 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
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 \( ( 1 - T + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
53 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
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 \( ( 1 - T )^{12}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
97 \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \)
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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

−2.73383667958780981315768664830, −2.69332083372566307677737546830, −2.69308418491491791606422836519, −2.65228847075675719087332906165, −2.30725466746086802692669425607, −2.28922229310592828992422862615, −2.28770458671375644732933588555, −2.26960219322207808899474211402, −2.22651257026645789001742480071, −2.14729078263155866334391724289, −2.04633903478552806248160723949, −1.99717167173994205111241387422, −1.85987817667794130010714958822, −1.74156224330484208101339782949, −1.65274340461044070660476508905, −1.35041444390008464398325110291, −1.17146700370922305512253497081, −1.09343449014481786752552608207, −0.996161358350229359359602826179, −0.988205427658800377376305394646, −0.863026540937325976582703561950, −0.850194197942549110536544032875, −0.77017596678406494857937505762, −0.63457102725617315966096467768, −0.52814996618226650826832024860, 0.52814996618226650826832024860, 0.63457102725617315966096467768, 0.77017596678406494857937505762, 0.850194197942549110536544032875, 0.863026540937325976582703561950, 0.988205427658800377376305394646, 0.996161358350229359359602826179, 1.09343449014481786752552608207, 1.17146700370922305512253497081, 1.35041444390008464398325110291, 1.65274340461044070660476508905, 1.74156224330484208101339782949, 1.85987817667794130010714958822, 1.99717167173994205111241387422, 2.04633903478552806248160723949, 2.14729078263155866334391724289, 2.22651257026645789001742480071, 2.26960219322207808899474211402, 2.28770458671375644732933588555, 2.28922229310592828992422862615, 2.30725466746086802692669425607, 2.65228847075675719087332906165, 2.69308418491491791606422836519, 2.69332083372566307677737546830, 2.73383667958780981315768664830

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

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