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Properties

Label 24-791e12-1.1-c0e12-0-0
Degree $24$
Conductor $6.000\times 10^{34}$
Sign $1$
Analytic cond. $1.43219\times 10^{-5}$
Root an. cond. $0.628299$
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  − 5·4-s − 2·7-s + 15·16-s − 2·23-s + 10·28-s − 2·29-s − 2·37-s + 2·43-s + 49-s + 4·53-s − 35·64-s + 2·67-s + 2·71-s + 2·79-s + 81-s + 10·92-s − 12·107-s − 30·112-s − 2·113-s + 10·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 10·148-s + 149-s + ⋯
L(s)  = 1  − 5·4-s − 2·7-s + 15·16-s − 2·23-s + 10·28-s − 2·29-s − 2·37-s + 2·43-s + 49-s + 4·53-s − 35·64-s + 2·67-s + 2·71-s + 2·79-s + 81-s + 10·92-s − 12·107-s − 30·112-s − 2·113-s + 10·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 10·148-s + 149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 113^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 113^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(7^{12} \cdot 113^{12}\)
Sign: $1$
Analytic conductor: \(1.43219\times 10^{-5}\)
Root analytic conductor: \(0.628299\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 7^{12} \cdot 113^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001418338092\)
\(L(\frac12)\) \(\approx\) \(0.001418338092\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
113 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
good2 \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
5 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
17 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
19 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
23 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
47 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4} \)
59 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
73 \( ( 1 + T^{4} )^{6} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
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 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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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

−3.71980327707765271699251803048, −3.66975026455840992415552781272, −3.50967685068487452495231272386, −3.38701033701332887080175762484, −3.31841413357012705846915034285, −3.25658011486191579791105560026, −3.19195062323345409122672725939, −2.99269709141153601052012192419, −2.92108509027432489732406483477, −2.72631301205427954576309583897, −2.47074174615031804079010987122, −2.42034104309377886082509403127, −2.39073263612703934979183172428, −2.38309149146194678000824939509, −2.29018704340616531511241849411, −2.10051497848160972054234793925, −1.96756945701567131966954028661, −1.69724643302410716621258612141, −1.34947849400939360524550330347, −1.27287488453217853658926950575, −1.27163810891445426735823538129, −1.22561629346740439755162228060, −1.00171147182685974879987293577, −0.833498245506992165125588121807, −0.06098023281085999802369342617, 0.06098023281085999802369342617, 0.833498245506992165125588121807, 1.00171147182685974879987293577, 1.22561629346740439755162228060, 1.27163810891445426735823538129, 1.27287488453217853658926950575, 1.34947849400939360524550330347, 1.69724643302410716621258612141, 1.96756945701567131966954028661, 2.10051497848160972054234793925, 2.29018704340616531511241849411, 2.38309149146194678000824939509, 2.39073263612703934979183172428, 2.42034104309377886082509403127, 2.47074174615031804079010987122, 2.72631301205427954576309583897, 2.92108509027432489732406483477, 2.99269709141153601052012192419, 3.19195062323345409122672725939, 3.25658011486191579791105560026, 3.31841413357012705846915034285, 3.38701033701332887080175762484, 3.50967685068487452495231272386, 3.66975026455840992415552781272, 3.71980327707765271699251803048

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

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