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Properties

Label 24-344e12-1.1-c0e12-0-0
Degree $24$
Conductor $2.746\times 10^{30}$
Sign $1$
Analytic cond. $6.55521\times 10^{-10}$
Root an. cond. $0.414340$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 4-s − 4·6-s + 3·9-s + 2·11-s + 2·12-s − 17-s − 6·18-s − 8·19-s − 4·22-s + 25-s + 2·27-s + 4·33-s + 2·34-s + 3·36-s + 16·38-s + 2·41-s + 43-s + 2·44-s − 6·49-s − 2·50-s − 2·51-s − 4·54-s − 16·57-s + 2·59-s − 8·66-s + ⋯
L(s)  = 1  − 2·2-s + 2·3-s + 4-s − 4·6-s + 3·9-s + 2·11-s + 2·12-s − 17-s − 6·18-s − 8·19-s − 4·22-s + 25-s + 2·27-s + 4·33-s + 2·34-s + 3·36-s + 16·38-s + 2·41-s + 43-s + 2·44-s − 6·49-s − 2·50-s − 2·51-s − 4·54-s − 16·57-s + 2·59-s − 8·66-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{36} \cdot 43^{12}\)
Sign: $1$
Analytic conductor: \(6.55521\times 10^{-10}\)
Root analytic conductor: \(0.414340\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{36} \cdot 43^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03343124986\)
\(L(\frac12)\) \(\approx\) \(0.03343124986\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
43 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
good3 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
5 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
7 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
11 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
19 \( ( 1 + T + T^{2} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
31 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
37 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
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 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
83 \( ( 1 + T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
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

−4.17012437001740917562665708441, −4.16837833795055493474315415483, −4.11668124125431580184409509170, −4.06355247702662862885831184033, −3.81701692849630725862921154122, −3.72577290809194607888574344161, −3.58127148465582222165267892519, −3.35317077862772509040807289345, −3.35063158084632041415262270894, −3.15147958605964482868145086568, −3.09765643261682192789617484897, −2.93243827019173581439980870553, −2.83840507886494907723339930088, −2.60738410633270137491027939155, −2.58329148140085896195049205708, −2.37009335252423658160639309270, −2.22998636931706005976451631392, −2.12113373965743263676717660908, −1.98239415132721060239527734989, −1.85878543382695606074049639385, −1.81894112642662739002343244860, −1.76980334769655229327833394960, −1.45210207811279877593634646987, −1.28930639302093755764573374300, −1.02944356236357182026139025103, 1.02944356236357182026139025103, 1.28930639302093755764573374300, 1.45210207811279877593634646987, 1.76980334769655229327833394960, 1.81894112642662739002343244860, 1.85878543382695606074049639385, 1.98239415132721060239527734989, 2.12113373965743263676717660908, 2.22998636931706005976451631392, 2.37009335252423658160639309270, 2.58329148140085896195049205708, 2.60738410633270137491027939155, 2.83840507886494907723339930088, 2.93243827019173581439980870553, 3.09765643261682192789617484897, 3.15147958605964482868145086568, 3.35063158084632041415262270894, 3.35317077862772509040807289345, 3.58127148465582222165267892519, 3.72577290809194607888574344161, 3.81701692849630725862921154122, 4.06355247702662862885831184033, 4.11668124125431580184409509170, 4.16837833795055493474315415483, 4.17012437001740917562665708441

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

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