Properties

Label 24-1216e12-1.1-c0e12-0-0
Degree $24$
Conductor $1.045\times 10^{37}$
Sign $1$
Analytic cond. $0.00249510$
Root an. cond. $0.779014$
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  + 3·9-s − 6·41-s + 6·49-s − 12·73-s + 3·81-s − 6·97-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 + 233-s + ⋯
L(s)  = 1  + 3·9-s − 6·41-s + 6·49-s − 12·73-s + 3·81-s − 6·97-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 + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{72} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(0.00249510\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{72} \cdot 19^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4859253585\)
\(L(\frac12)\) \(\approx\) \(0.4859253585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.16764822558467111678555864723, −3.13037903922762753027582431588, −3.12674812637773752249467569328, −3.06482814512033183405177947461, −3.00960700019456537680335124638, −2.92694624277638073619558766126, −2.92200061633886908781910854403, −2.55060685991914191454508305256, −2.47497633780502933382292621077, −2.44055951308115548738215604526, −2.42113612990466492892477293578, −2.41509851765868934369956338044, −2.25824319032103595698018379715, −2.12104111770290365155101052704, −1.71730106099862639184364120584, −1.69335723270342150790973741657, −1.62152191060400622263789908734, −1.55430523034076738506449401481, −1.52592928334195601004677255390, −1.40818187545204399092170875035, −1.29478865325345599634721666383, −1.23910926268359474326035321563, −1.13939779746055201226335974365, −0.874569412338624668585024776104, −0.31488439567298742981317308735, 0.31488439567298742981317308735, 0.874569412338624668585024776104, 1.13939779746055201226335974365, 1.23910926268359474326035321563, 1.29478865325345599634721666383, 1.40818187545204399092170875035, 1.52592928334195601004677255390, 1.55430523034076738506449401481, 1.62152191060400622263789908734, 1.69335723270342150790973741657, 1.71730106099862639184364120584, 2.12104111770290365155101052704, 2.25824319032103595698018379715, 2.41509851765868934369956338044, 2.42113612990466492892477293578, 2.44055951308115548738215604526, 2.47497633780502933382292621077, 2.55060685991914191454508305256, 2.92200061633886908781910854403, 2.92694624277638073619558766126, 3.00960700019456537680335124638, 3.06482814512033183405177947461, 3.12674812637773752249467569328, 3.13037903922762753027582431588, 3.16764822558467111678555864723

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

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