Properties

Label 24-740e12-1.1-c0e12-0-1
Degree $24$
Conductor $2.696\times 10^{34}$
Sign $1$
Analytic cond. $6.43674\times 10^{-6}$
Root an. cond. $0.607707$
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  + 2·8-s − 6·41-s − 12·61-s + 64-s − 6·73-s + 6·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 2·8-s − 6·41-s − 12·61-s + 64-s − 6·73-s + 6·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 5^{12} \cdot 37^{12}\)
Sign: $1$
Analytic conductor: \(6.43674\times 10^{-6}\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{740} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 5^{12} \cdot 37^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3064278652\)
\(L(\frac12)\) \(\approx\) \(0.3064278652\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.51922274753762514892397980423, −3.47161774006450825583588754456, −3.43660816832380720117629417738, −3.26184026943610385532576172922, −3.16940710556418082890194802028, −3.07539475163792547946105972730, −3.03684557946812512029097166275, −3.00050363484046691961349481278, −2.96681040535795465259416308661, −2.87569636221968225739938126366, −2.77692594114867951700221673542, −2.69465447410502764290323074582, −2.37092716183250772645325861075, −2.07598423846584166328862841398, −1.97992667850293371991947462438, −1.92502005866318154057584966257, −1.90368829215810200105774943686, −1.78805338561687805984380613003, −1.68215166914304830762483112233, −1.59892558177097611711710058425, −1.47010136264303606411520705904, −1.44385222178337068836010524426, −1.16952663504902888006865938130, −1.10294812559639198994348741592, −0.52857870346125036296112758498, 0.52857870346125036296112758498, 1.10294812559639198994348741592, 1.16952663504902888006865938130, 1.44385222178337068836010524426, 1.47010136264303606411520705904, 1.59892558177097611711710058425, 1.68215166914304830762483112233, 1.78805338561687805984380613003, 1.90368829215810200105774943686, 1.92502005866318154057584966257, 1.97992667850293371991947462438, 2.07598423846584166328862841398, 2.37092716183250772645325861075, 2.69465447410502764290323074582, 2.77692594114867951700221673542, 2.87569636221968225739938126366, 2.96681040535795465259416308661, 3.00050363484046691961349481278, 3.03684557946812512029097166275, 3.07539475163792547946105972730, 3.16940710556418082890194802028, 3.26184026943610385532576172922, 3.43660816832380720117629417738, 3.47161774006450825583588754456, 3.51922274753762514892397980423

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

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