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Properties

Label 24-38e24-1.1-c0e12-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.007041436620\)
\(L(\frac12)\) \(\approx\) \(0.007041436620\)
\(L(1)\) 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

−3.19273433209094934220494331645, −3.15829891789116712813898812895, −3.12372278887927279642290232269, −3.09880078526098251944154280990, −3.04422183801025061771193145518, −2.86051692980694759961435183125, −2.62435094381009984568625753511, −2.57948173831627851045723270462, −2.51019015334262736459366032698, −2.40903223547916532589738432403, −2.29712804652497751872714700174, −2.18894092027589159500219909049, −2.18416526293137235012767347832, −2.13188352626739898071159879540, −1.79808480386251094680648567362, −1.61157676859970763923787486350, −1.58359481563180506968529824179, −1.53716816421469704562920067556, −1.50453479921146156644372444182, −1.39483876637080032541160855253, −1.37806062641709026246631063626, −1.08984087481933387496895294745, −0.912850641910967865228431253617, −0.51461945637253604653206781229, −0.04721909344781014583097696863, 0.04721909344781014583097696863, 0.51461945637253604653206781229, 0.912850641910967865228431253617, 1.08984087481933387496895294745, 1.37806062641709026246631063626, 1.39483876637080032541160855253, 1.50453479921146156644372444182, 1.53716816421469704562920067556, 1.58359481563180506968529824179, 1.61157676859970763923787486350, 1.79808480386251094680648567362, 2.13188352626739898071159879540, 2.18416526293137235012767347832, 2.18894092027589159500219909049, 2.29712804652497751872714700174, 2.40903223547916532589738432403, 2.51019015334262736459366032698, 2.57948173831627851045723270462, 2.62435094381009984568625753511, 2.86051692980694759961435183125, 3.04422183801025061771193145518, 3.09880078526098251944154280990, 3.12372278887927279642290232269, 3.15829891789116712813898812895, 3.19273433209094934220494331645

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

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