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Properties

Label 24-38e24-1.1-c0e12-0-2
Degree $24$
Conductor $8.219\times 10^{37}$
Sign $1$
Analytic cond. $0.0196196$
Root an. cond. $0.848910$
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 + 6·37-s − 6·49-s + 64-s + 6·113-s − 6·121-s + 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 + 6·37-s − 6·49-s + 64-s + 6·113-s − 6·121-s + 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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 19^{24}\)
Sign: $1$
Analytic conductor: \(0.0196196\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 19^{24} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.627559459\)
\(L(\frac12)\) \(\approx\) \(1.627559459\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{3} + T^{6} )^{2} \)
19 \( 1 \)
good3 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
5 \( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} \)
7 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
11 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
13 \( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} \)
17 \( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} \)
23 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
29 \( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} \)
31 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{6} \)
41 \( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} \)
43 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
53 \( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} \)
59 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
61 \( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} \)
67 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
73 \( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} \)
79 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
89 \( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} \)
97 \( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} \)
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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

−3.22548490183505783395225855370, −3.11369423793324844311584924725, −3.08025006432274877124035186008, −3.07600560339644968896983209855, −2.79088688570647037966452049275, −2.71126620787086906439744874980, −2.63903671755611570237179309014, −2.60288029160539535657622058719, −2.51360768531416406287295346914, −2.43922853860226976871691112071, −2.41574862324661221882492003024, −2.17575970016404424170055900549, −2.13603889831888095821234411746, −1.97397212393113611542562201465, −1.73003146783045559097353732961, −1.72375609122142383941560053139, −1.61205313937249591726604730771, −1.52120590255491963475897268487, −1.48008503117779649418788219783, −1.44165963038130625532282986764, −1.33185650954271832206818610862, −0.915433016892156785637691307476, −0.801908867278651279463621912622, −0.76504225728351828022175193910, −0.69204875077048888336504983668, 0.69204875077048888336504983668, 0.76504225728351828022175193910, 0.801908867278651279463621912622, 0.915433016892156785637691307476, 1.33185650954271832206818610862, 1.44165963038130625532282986764, 1.48008503117779649418788219783, 1.52120590255491963475897268487, 1.61205313937249591726604730771, 1.72375609122142383941560053139, 1.73003146783045559097353732961, 1.97397212393113611542562201465, 2.13603889831888095821234411746, 2.17575970016404424170055900549, 2.41574862324661221882492003024, 2.43922853860226976871691112071, 2.51360768531416406287295346914, 2.60288029160539535657622058719, 2.63903671755611570237179309014, 2.71126620787086906439744874980, 2.79088688570647037966452049275, 3.07600560339644968896983209855, 3.08025006432274877124035186008, 3.11369423793324844311584924725, 3.22548490183505783395225855370

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

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