Properties

Label 24-384e12-1.1-c5e12-0-0
Degree $24$
Conductor $1.028\times 10^{31}$
Sign $1$
Analytic cond. $2.97783\times 10^{21}$
Root an. cond. $7.84776$
Motivic weight $5$
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  − 486·9-s − 4.88e3·17-s + 9.42e3·25-s − 2.50e4·41-s − 3.30e4·49-s − 1.87e5·73-s + 1.37e5·81-s − 5.75e5·89-s + 9.38e4·97-s − 1.45e5·113-s + 6.89e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.37e6·153-s + 157-s + 163-s + 167-s + 3.49e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·9-s − 4.10·17-s + 3.01·25-s − 2.33·41-s − 1.96·49-s − 4.10·73-s + 7/3·81-s − 7.70·89-s + 1.01·97-s − 1.07·113-s + 4.28·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 8.20·153-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 9.42·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{84} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(2.97783\times 10^{21}\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{84} \cdot 3^{12} ,\ ( \ : [5/2]^{12} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.04774230993\)
\(L(\frac12)\) \(\approx\) \(0.04774230993\)
\(L(\frac{7}{2})\) 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 \( ( 1 + p^{4} T^{2} )^{6} \)
good5 \( ( 1 - 942 p T^{2} + 1474167 T^{4} + 32638729228 T^{6} + 1474167 p^{10} T^{8} - 942 p^{21} T^{10} + p^{30} T^{12} )^{2} \)
7 \( ( 1 + 16530 T^{2} + 170506047 T^{4} + 1177445256636 T^{6} + 170506047 p^{10} T^{8} + 16530 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
11 \( ( 1 - 344946 T^{2} + 82897405815 T^{4} - 13915185353972124 T^{6} + 82897405815 p^{10} T^{8} - 344946 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
13 \( ( 1 - 1749198 T^{2} + 1426570744647 T^{4} - 676280623930934820 T^{6} + 1426570744647 p^{10} T^{8} - 1749198 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
17 \( ( 1 + 1222 T + 1251839 T^{2} + 497295444 T^{3} + 1251839 p^{5} T^{4} + 1222 p^{10} T^{5} + p^{15} T^{6} )^{4} \)
19 \( ( 1 - 4229730 T^{2} + 16355477959623 T^{4} - 51098128943446652668 T^{6} + 16355477959623 p^{10} T^{8} - 4229730 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
23 \( ( 1 + 22472298 T^{2} + 227430925696575 T^{4} + \)\(15\!\cdots\!32\)\( T^{6} + 227430925696575 p^{10} T^{8} + 22472298 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
29 \( ( 1 - 45734454 T^{2} + 1857413971003527 T^{4} - \)\(40\!\cdots\!48\)\( T^{6} + 1857413971003527 p^{10} T^{8} - 45734454 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
31 \( ( 1 + 77752482 T^{2} + 2614361277946479 T^{4} + \)\(65\!\cdots\!84\)\( T^{6} + 2614361277946479 p^{10} T^{8} + 77752482 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
37 \( ( 1 - 232347390 T^{2} + 21316181790168663 T^{4} - \)\(13\!\cdots\!48\)\( T^{6} + 21316181790168663 p^{10} T^{8} - 232347390 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
41 \( ( 1 + 6274 T + 244052615 T^{2} + 893536860348 T^{3} + 244052615 p^{5} T^{4} + 6274 p^{10} T^{5} + p^{15} T^{6} )^{4} \)
43 \( ( 1 - 483419058 T^{2} + 123803325975021495 T^{4} - \)\(21\!\cdots\!96\)\( T^{6} + 123803325975021495 p^{10} T^{8} - 483419058 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
47 \( ( 1 + 393125946 T^{2} + 155047382878218351 T^{4} + \)\(40\!\cdots\!40\)\( T^{6} + 155047382878218351 p^{10} T^{8} + 393125946 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
53 \( ( 1 - 2023802886 T^{2} + 1873845878937911319 T^{4} - \)\(10\!\cdots\!72\)\( T^{6} + 1873845878937911319 p^{10} T^{8} - 2023802886 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
59 \( ( 1 - 19398678 p T^{2} + 1913304132589469655 T^{4} - \)\(12\!\cdots\!08\)\( T^{6} + 1913304132589469655 p^{10} T^{8} - 19398678 p^{21} T^{10} + p^{30} T^{12} )^{2} \)
61 \( ( 1 - 3538610766 T^{2} + 5894581151789944167 T^{4} - \)\(60\!\cdots\!56\)\( T^{6} + 5894581151789944167 p^{10} T^{8} - 3538610766 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
67 \( ( 1 - 5431347714 T^{2} + 14867265317327351079 T^{4} - \)\(25\!\cdots\!44\)\( T^{6} + 14867265317327351079 p^{10} T^{8} - 5431347714 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
71 \( ( 1 + 6495534090 T^{2} + 18049795256179945503 T^{4} + \)\(34\!\cdots\!80\)\( T^{6} + 18049795256179945503 p^{10} T^{8} + 6495534090 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
73 \( ( 1 + 46758 T + 6179984967 T^{2} + 181584728514740 T^{3} + 6179984967 p^{5} T^{4} + 46758 p^{10} T^{5} + p^{15} T^{6} )^{4} \)
79 \( ( 1 + 14190311490 T^{2} + 94960210564756835151 T^{4} + \)\(37\!\cdots\!24\)\( T^{6} + 94960210564756835151 p^{10} T^{8} + 14190311490 p^{20} T^{10} + p^{30} T^{12} )^{2} \)
83 \( ( 1 - 125814870 p T^{2} + 52267541353483961415 T^{4} - \)\(20\!\cdots\!60\)\( T^{6} + 52267541353483961415 p^{10} T^{8} - 125814870 p^{21} T^{10} + p^{30} T^{12} )^{2} \)
89 \( ( 1 + 143886 T + 20607412119 T^{2} + 1635794835676644 T^{3} + 20607412119 p^{5} T^{4} + 143886 p^{10} T^{5} + p^{15} T^{6} )^{4} \)
97 \( ( 1 - 23466 T + 8123474415 T^{2} - 995941183942028 T^{3} + 8123474415 p^{5} T^{4} - 23466 p^{10} T^{5} + p^{15} T^{6} )^{4} \)
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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

−2.94365877847732722086851195070, −2.94029617897843226032839702022, −2.74033075021766116724948194045, −2.54292538526697547812994678260, −2.43265586306210158128717190850, −2.31512082627375235329583057311, −2.20636083585629434823315044468, −2.18661829118034553055665285950, −2.04069676201246267082796439914, −1.82990566131618937362011109679, −1.77339007071295132442182071968, −1.71972848197837880656052869215, −1.66908964185988801303935278232, −1.46745030330803608280754343788, −1.29630723101957361365019757221, −1.26255104526552920727659174361, −0.994199208038070722862142231870, −0.866086013749546980511020047501, −0.809355370337969464397926778370, −0.62309427377989873352467210785, −0.58521615543133923623183863939, −0.41165996980457245095582716080, −0.12344515772578131591357926600, −0.087057269149215143551908486989, −0.06496295618053587088206836800, 0.06496295618053587088206836800, 0.087057269149215143551908486989, 0.12344515772578131591357926600, 0.41165996980457245095582716080, 0.58521615543133923623183863939, 0.62309427377989873352467210785, 0.809355370337969464397926778370, 0.866086013749546980511020047501, 0.994199208038070722862142231870, 1.26255104526552920727659174361, 1.29630723101957361365019757221, 1.46745030330803608280754343788, 1.66908964185988801303935278232, 1.71972848197837880656052869215, 1.77339007071295132442182071968, 1.82990566131618937362011109679, 2.04069676201246267082796439914, 2.18661829118034553055665285950, 2.20636083585629434823315044468, 2.31512082627375235329583057311, 2.43265586306210158128717190850, 2.54292538526697547812994678260, 2.74033075021766116724948194045, 2.94029617897843226032839702022, 2.94365877847732722086851195070

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

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