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Properties

Label 24-580e12-1.1-c0e12-0-1
Degree $24$
Conductor $1.449\times 10^{33}$
Sign $1$
Analytic cond. $3.45956\times 10^{-7}$
Root an. cond. $0.538012$
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 − 2·9-s − 2·13-s + 25-s − 2·36-s + 4·37-s − 2·41-s − 2·52-s − 12·53-s + 2·61-s + 81-s + 2·89-s + 100-s + 2·101-s + 4·117-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s + 2·169-s + ⋯
L(s)  = 1  + 4-s − 2·9-s − 2·13-s + 25-s − 2·36-s + 4·37-s − 2·41-s − 2·52-s − 12·53-s + 2·61-s + 81-s + 2·89-s + 100-s + 2·101-s + 4·117-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 5^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(3.45956\times 10^{-7}\)
Root analytic conductor: \(0.538012\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 5^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1764203791\)
\(L(\frac12)\) \(\approx\) \(0.1764203791\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.83778837543750695991056288713, −3.62709277820225461534789473539, −3.53686931522125783347686673769, −3.51879988002988836601401239455, −3.49190023963882617646859137700, −3.26755488682648090799679381197, −3.05350803801240837879888378145, −2.96653439068943675524219499991, −2.83562131975765253395632386472, −2.82826125297821486258791941414, −2.74907721805440472559588737103, −2.67328655537037388072423104187, −2.64165103612795998003154023657, −2.61897881850716786709465327375, −2.47849300234379852912681074900, −2.26384044545561952327919457966, −1.88642398279023037557155936738, −1.88001805059847198665652811014, −1.83745610814302082291887939496, −1.69134475124851163075041598457, −1.61442408924215957215912133969, −1.40937526869462247659337287049, −1.35833975393823302682912603802, −0.957700839677205841569781250180, −0.66035084072691818627881540098, 0.66035084072691818627881540098, 0.957700839677205841569781250180, 1.35833975393823302682912603802, 1.40937526869462247659337287049, 1.61442408924215957215912133969, 1.69134475124851163075041598457, 1.83745610814302082291887939496, 1.88001805059847198665652811014, 1.88642398279023037557155936738, 2.26384044545561952327919457966, 2.47849300234379852912681074900, 2.61897881850716786709465327375, 2.64165103612795998003154023657, 2.67328655537037388072423104187, 2.74907721805440472559588737103, 2.82826125297821486258791941414, 2.83562131975765253395632386472, 2.96653439068943675524219499991, 3.05350803801240837879888378145, 3.26755488682648090799679381197, 3.49190023963882617646859137700, 3.51879988002988836601401239455, 3.53686931522125783347686673769, 3.62709277820225461534789473539, 3.83778837543750695991056288713

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

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