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Properties

Label 24-3248e12-1.1-c0e12-0-2
Degree $24$
Conductor $1.378\times 10^{42}$
Sign $1$
Analytic cond. $329.062$
Root an. cond. $1.27317$
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  + 4-s − 2·7-s − 2·9-s − 14·11-s − 2·28-s − 2·36-s − 14·44-s + 49-s − 2·53-s + 4·63-s + 2·67-s + 28·77-s + 12·79-s + 81-s + 28·99-s + 2·107-s + 2·109-s + 2·113-s + 103·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 4-s − 2·7-s − 2·9-s − 14·11-s − 2·28-s − 2·36-s − 14·44-s + 49-s − 2·53-s + 4·63-s + 2·67-s + 28·77-s + 12·79-s + 81-s + 28·99-s + 2·107-s + 2·109-s + 2·113-s + 103·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 7^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(329.062\)
Root analytic conductor: \(1.27317\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 7^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03720829939\)
\(L(\frac12)\) \(\approx\) \(0.03720829939\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
good3 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
5 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
11 \( ( 1 + T )^{12}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
13 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
17 \( ( 1 + T^{4} )^{6} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
23 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
31 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
41 \( ( 1 + T^{4} )^{6} \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
47 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
59 \( ( 1 + T^{4} )^{6} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
73 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
79 \( ( 1 - T )^{12}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
89 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
97 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
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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

−2.80619435044502337628390718237, −2.75816295166202988366511147127, −2.75121595906662289882376630550, −2.74472617102369417241371299980, −2.42974730887494632932217803472, −2.40147425749081777994261807526, −2.31781084987927365357351847820, −2.30983337006509830862908816689, −2.25137265729258605621425645057, −2.20289543551264693315129232256, −2.18810645747786248825370074752, −2.00265048721836501657868748366, −1.98793017048808606655934558805, −1.91583934230900068236252894528, −1.87870359576323903774406765781, −1.73207396799326423371302566508, −1.61527636418388386204893832755, −1.16780635880960428453889370481, −0.922971012896105521524703967037, −0.865696155527756897274513331254, −0.800471469547880981632014139605, −0.58928495555956520307320991661, −0.57839118127592281412143114182, −0.32841126811539604059450013499, −0.13953630122734436417359568789, 0.13953630122734436417359568789, 0.32841126811539604059450013499, 0.57839118127592281412143114182, 0.58928495555956520307320991661, 0.800471469547880981632014139605, 0.865696155527756897274513331254, 0.922971012896105521524703967037, 1.16780635880960428453889370481, 1.61527636418388386204893832755, 1.73207396799326423371302566508, 1.87870359576323903774406765781, 1.91583934230900068236252894528, 1.98793017048808606655934558805, 2.00265048721836501657868748366, 2.18810645747786248825370074752, 2.20289543551264693315129232256, 2.25137265729258605621425645057, 2.30983337006509830862908816689, 2.31781084987927365357351847820, 2.40147425749081777994261807526, 2.42974730887494632932217803472, 2.74472617102369417241371299980, 2.75121595906662289882376630550, 2.75816295166202988366511147127, 2.80619435044502337628390718237

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

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