Properties

Label 24-384e12-1.1-c1e12-0-1
Degree $24$
Conductor $1.028\times 10^{31}$
Sign $1$
Analytic cond. $690716.$
Root an. cond. $1.75107$
Motivic weight $1$
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  + 2·3-s − 8·7-s + 2·9-s + 4·13-s + 12·19-s − 16·21-s − 2·27-s + 4·37-s + 8·39-s − 12·43-s − 20·49-s + 24·57-s − 12·61-s − 16·63-s − 28·67-s − 7·81-s − 32·91-s − 8·97-s − 56·103-s − 12·109-s + 8·111-s + 8·117-s + 127-s − 24·129-s + 131-s − 96·133-s + 137-s + ⋯
L(s)  = 1  + 1.15·3-s − 3.02·7-s + 2/3·9-s + 1.10·13-s + 2.75·19-s − 3.49·21-s − 0.384·27-s + 0.657·37-s + 1.28·39-s − 1.82·43-s − 2.85·49-s + 3.17·57-s − 1.53·61-s − 2.01·63-s − 3.42·67-s − 7/9·81-s − 3.35·91-s − 0.812·97-s − 5.51·103-s − 1.14·109-s + 0.759·111-s + 0.739·117-s + 0.0887·127-s − 2.11·129-s + 0.0873·131-s − 8.32·133-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{84} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(690716.\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{84} \cdot 3^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.966077971\)
\(L(\frac12)\) \(\approx\) \(1.966077971\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 2 T + 2 T^{2} + 2 T^{3} - 5 T^{4} + 20 T^{5} - 28 T^{6} + 20 p T^{7} - 5 p^{2} T^{8} + 2 p^{3} T^{9} + 2 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
good5 \( 1 - 6 p T^{4} - 49 T^{8} + 12796 T^{12} - 49 p^{4} T^{16} - 6 p^{9} T^{20} + p^{12} T^{24} \)
7 \( ( 1 + 2 T + 15 T^{2} + 20 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
11 \( 1 - 62 T^{4} + 11023 T^{8} - 2631620 T^{12} + 11023 p^{4} T^{16} - 62 p^{8} T^{20} + p^{12} T^{24} \)
13 \( ( 1 - 2 T + 2 T^{2} + 6 T^{3} - 25 T^{4} - 412 T^{5} + 892 T^{6} - 412 p T^{7} - 25 p^{2} T^{8} + 6 p^{3} T^{9} + 2 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
17 \( ( 1 - 62 T^{2} + 1903 T^{4} - 38180 T^{6} + 1903 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 - 6 T + 18 T^{2} - 82 T^{3} + 539 T^{4} - 2636 T^{5} + 9476 T^{6} - 2636 p T^{7} + 539 p^{2} T^{8} - 82 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( ( 1 - 86 T^{2} + 3791 T^{4} - 105684 T^{6} + 3791 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( 1 - 830 T^{4} + 2253679 T^{8} - 1165110596 T^{12} + 2253679 p^{4} T^{16} - 830 p^{8} T^{20} + p^{12} T^{24} \)
31 \( ( 1 - 150 T^{2} + 10019 T^{4} - 392444 T^{6} + 10019 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 2 T + 2 T^{2} + 54 T^{3} + 567 T^{4} - 8764 T^{5} + 17852 T^{6} - 8764 p T^{7} + 567 p^{2} T^{8} + 54 p^{3} T^{9} + 2 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( ( 1 + 138 T^{2} + 7887 T^{4} + 320492 T^{6} + 7887 p^{2} T^{8} + 138 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 + 6 T + 18 T^{2} + 226 T^{3} + 4235 T^{4} + 18188 T^{5} + 58436 T^{6} + 18188 p T^{7} + 4235 p^{2} T^{8} + 226 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( ( 1 + 170 T^{2} + 15791 T^{4} + 908172 T^{6} + 15791 p^{2} T^{8} + 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( 1 + 7714 T^{4} + 19237903 T^{8} + 30633057916 T^{12} + 19237903 p^{4} T^{16} + 7714 p^{8} T^{20} + p^{12} T^{24} \)
59 \( ( 1 - 30 T + 450 T^{2} - 3458 T^{3} + 3915 T^{4} + 231740 T^{5} - 2735068 T^{6} + 231740 p T^{7} + 3915 p^{2} T^{8} - 3458 p^{3} T^{9} + 450 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )( 1 + 30 T + 450 T^{2} + 3458 T^{3} + 3915 T^{4} - 231740 T^{5} - 2735068 T^{6} - 231740 p T^{7} + 3915 p^{2} T^{8} + 3458 p^{3} T^{9} + 450 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} ) \)
61 \( ( 1 + 6 T + 18 T^{2} + 430 T^{3} - 121 T^{4} - 30796 T^{5} - 90148 T^{6} - 30796 p T^{7} - 121 p^{2} T^{8} + 430 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( ( 1 + 14 T + 98 T^{2} + 706 T^{3} + 9435 T^{4} + 112164 T^{5} + 894884 T^{6} + 112164 p T^{7} + 9435 p^{2} T^{8} + 706 p^{3} T^{9} + 98 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 230 T^{2} + 30127 T^{4} - 2527028 T^{6} + 30127 p^{2} T^{8} - 230 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 166 T^{2} + 19007 T^{4} - 1414164 T^{6} + 19007 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 358 T^{2} + 58915 T^{4} - 5817628 T^{6} + 58915 p^{2} T^{8} - 358 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( 1 - 1374 T^{4} + 18563631 T^{8} - 336062521604 T^{12} + 18563631 p^{4} T^{16} - 1374 p^{8} T^{20} + p^{12} T^{24} \)
89 \( ( 1 + 322 T^{2} + 51919 T^{4} + 5548348 T^{6} + 51919 p^{2} T^{8} + 322 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( ( 1 + 2 T + 163 T^{2} - 220 T^{3} + 163 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
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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

−3.77104349573043568555564954477, −3.61188680304854987620891507409, −3.55802835446090136128909585342, −3.42116638946442823399670210636, −3.40432520046975422127802405606, −3.31327204217960823529705412766, −3.11146926724121370900706633168, −3.04602305193807420428062307323, −2.93724514502245192657080780505, −2.84114913044309593773789383027, −2.73759855601906288781256491800, −2.69672537317780218443155911242, −2.69456275476230197630858469209, −2.42684058434564036129609477098, −2.42500442209953750697873536026, −1.80122298426281180647347900878, −1.69902914385269716382626458823, −1.62264120640177571896522724276, −1.61553805336370046337646677342, −1.51842017909992403566791883762, −1.50364311978344121610032969637, −1.27117601779801645650005727297, −0.57960252942503700806024779729, −0.56065310157211185376996434404, −0.25408391479250120866410148931, 0.25408391479250120866410148931, 0.56065310157211185376996434404, 0.57960252942503700806024779729, 1.27117601779801645650005727297, 1.50364311978344121610032969637, 1.51842017909992403566791883762, 1.61553805336370046337646677342, 1.62264120640177571896522724276, 1.69902914385269716382626458823, 1.80122298426281180647347900878, 2.42500442209953750697873536026, 2.42684058434564036129609477098, 2.69456275476230197630858469209, 2.69672537317780218443155911242, 2.73759855601906288781256491800, 2.84114913044309593773789383027, 2.93724514502245192657080780505, 3.04602305193807420428062307323, 3.11146926724121370900706633168, 3.31327204217960823529705412766, 3.40432520046975422127802405606, 3.42116638946442823399670210636, 3.55802835446090136128909585342, 3.61188680304854987620891507409, 3.77104349573043568555564954477

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

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