Properties

Label 24-696e12-1.1-c0e12-0-0
Degree $24$
Conductor $1.292\times 10^{34}$
Sign $1$
Analytic cond. $3.08458\times 10^{-6}$
Root an. cond. $0.589363$
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 + 4·7-s + 9-s − 5·25-s + 4·28-s + 36-s + 6·49-s + 4·63-s − 5·100-s − 4·103-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 20·175-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 4-s + 4·7-s + 9-s − 5·25-s + 4·28-s + 36-s + 6·49-s + 4·63-s − 5·100-s − 4·103-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 20·175-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{36} \cdot 3^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(3.08458\times 10^{-6}\)
Root analytic conductor: \(0.589363\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{696} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{36} \cdot 3^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7321805904\)
\(L(\frac12)\) \(\approx\) \(0.7321805904\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
good5 \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4} \)
11 \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
17 \( ( 1 + T^{2} )^{12} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
41 \( ( 1 + T^{2} )^{12} \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
89 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
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

−3.73378966120202576426189686017, −3.68532654378712613760343324325, −3.64033351988256503052774449853, −3.56519343140242491459884911453, −3.24987623437680097302514096422, −3.15553236382923089954882233405, −3.14604470173222484565518757208, −2.71397004237103688910898506345, −2.68312323027722634749871943907, −2.67598222601990769487530475632, −2.66448839684159538506584880580, −2.62433708015577470368727045502, −2.45873478060512447878353155825, −2.23493169068335331467866167531, −2.22852492742317947207293339778, −1.94807405121364322669598241939, −1.91528600805496693327730537925, −1.73040100975832816726026899371, −1.68916000890826963098331427563, −1.66232465840927454142185753147, −1.49692502341862439712507751498, −1.44735874157297990972715123671, −1.24049475532175447737442231730, −1.14972633776720953582138618312, −0.919520099622317390870606543184, 0.919520099622317390870606543184, 1.14972633776720953582138618312, 1.24049475532175447737442231730, 1.44735874157297990972715123671, 1.49692502341862439712507751498, 1.66232465840927454142185753147, 1.68916000890826963098331427563, 1.73040100975832816726026899371, 1.91528600805496693327730537925, 1.94807405121364322669598241939, 2.22852492742317947207293339778, 2.23493169068335331467866167531, 2.45873478060512447878353155825, 2.62433708015577470368727045502, 2.66448839684159538506584880580, 2.67598222601990769487530475632, 2.68312323027722634749871943907, 2.71397004237103688910898506345, 3.14604470173222484565518757208, 3.15553236382923089954882233405, 3.24987623437680097302514096422, 3.56519343140242491459884911453, 3.64033351988256503052774449853, 3.68532654378712613760343324325, 3.73378966120202576426189686017

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

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