Properties

Label 24-15e24-1.1-c3e12-0-0
Degree $24$
Conductor $1.683\times 10^{28}$
Sign $1$
Analytic cond. $2.99628\times 10^{13}$
Root an. cond. $3.64354$
Motivic weight $3$
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  − 24·7-s + 108·13-s − 18·16-s − 1.24e3·31-s − 828·37-s + 96·43-s + 288·49-s + 96·61-s − 1.63e3·67-s − 3.97e3·73-s − 2.59e3·91-s − 2.77e3·97-s − 840·103-s + 432·112-s + 7.28e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.83e3·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.29·7-s + 2.30·13-s − 0.281·16-s − 7.23·31-s − 3.67·37-s + 0.340·43-s + 0.839·49-s + 0.201·61-s − 2.97·67-s − 6.36·73-s − 2.98·91-s − 2.90·97-s − 0.803·103-s + 0.364·112-s + 5.47·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.65·169-s + 0.000439·173-s + 0.000417·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{24} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(2.99628\times 10^{13}\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{24} \cdot 5^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.1656058295\)
\(L(\frac12)\) \(\approx\) \(0.1656058295\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 9 p T^{4} - 2799 T^{8} - 991 p^{7} T^{12} - 2799 p^{12} T^{16} + 9 p^{25} T^{20} + p^{36} T^{24} \)
7 \( ( 1 + 12 T + 72 T^{2} + 1556 T^{3} + 22227 T^{4} + 690456 T^{5} + 7895696 T^{6} + 690456 p^{3} T^{7} + 22227 p^{6} T^{8} + 1556 p^{9} T^{9} + 72 p^{12} T^{10} + 12 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
11 \( ( 1 - 3642 T^{2} + 9286791 T^{4} - 14142283148 T^{6} + 9286791 p^{6} T^{8} - 3642 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
13 \( ( 1 - 54 T + 1458 T^{2} - 5038 T^{3} + 1205127 T^{4} - 333009972 T^{5} + 16238154044 T^{6} - 333009972 p^{3} T^{7} + 1205127 p^{6} T^{8} - 5038 p^{9} T^{9} + 1458 p^{12} T^{10} - 54 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
17 \( 1 - 49620282 T^{4} + 153074918028111 T^{8} + \)\(22\!\cdots\!32\)\( T^{12} + 153074918028111 p^{12} T^{16} - 49620282 p^{24} T^{20} + p^{36} T^{24} \)
19 \( ( 1 - 14514 T^{2} + 200815575 T^{4} - 1429853859740 T^{6} + 200815575 p^{6} T^{8} - 14514 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
23 \( 1 - 113635674 T^{4} + 25093184434350255 T^{8} - \)\(50\!\cdots\!20\)\( T^{12} + 25093184434350255 p^{12} T^{16} - 113635674 p^{24} T^{20} + p^{36} T^{24} \)
29 \( ( 1 + 87480 T^{2} + 3995901963 T^{4} + 116580057138160 T^{6} + 3995901963 p^{6} T^{8} + 87480 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
31 \( ( 1 + 312 T + 109101 T^{2} + 18922768 T^{3} + 109101 p^{3} T^{4} + 312 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
37 \( ( 1 + 414 T + 85698 T^{2} + 6372422 T^{3} + 2097524535 T^{4} + 1542455626884 T^{5} + 479126853002588 T^{6} + 1542455626884 p^{3} T^{7} + 2097524535 p^{6} T^{8} + 6372422 p^{9} T^{9} + 85698 p^{12} T^{10} + 414 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
41 \( ( 1 - 45840 T^{2} + 9577213203 T^{4} - 226162139163680 T^{6} + 9577213203 p^{6} T^{8} - 45840 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
43 \( ( 1 - 48 T + 1152 T^{2} + 5220464 T^{3} - 794630085 T^{4} - 1048734972768 T^{5} + 64881314738432 T^{6} - 1048734972768 p^{3} T^{7} - 794630085 p^{6} T^{8} + 5220464 p^{9} T^{9} + 1152 p^{12} T^{10} - 48 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
47 \( 1 - 10286977914 T^{4} + \)\(26\!\cdots\!55\)\( T^{8} - \)\(23\!\cdots\!20\)\( T^{12} + \)\(26\!\cdots\!55\)\( p^{12} T^{16} - 10286977914 p^{24} T^{20} + p^{36} T^{24} \)
53 \( 1 + 39371267766 T^{4} + \)\(15\!\cdots\!75\)\( T^{8} + \)\(36\!\cdots\!60\)\( T^{12} + \)\(15\!\cdots\!75\)\( p^{12} T^{16} + 39371267766 p^{24} T^{20} + p^{36} T^{24} \)
59 \( ( 1 + 811098 T^{2} + 339081833511 T^{4} + 86183215682837452 T^{6} + 339081833511 p^{6} T^{8} + 811098 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
61 \( ( 1 - 24 T + 339855 T^{2} - 80914160 T^{3} + 339855 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
67 \( ( 1 + 816 T + 332928 T^{2} - 80035312 T^{3} - 84748766325 T^{4} - 69675584129184 T^{5} - 25437213790895872 T^{6} - 69675584129184 p^{3} T^{7} - 84748766325 p^{6} T^{8} - 80035312 p^{9} T^{9} + 332928 p^{12} T^{10} + 816 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
71 \( ( 1 - 1544490 T^{2} + 1146403782783 T^{4} - 515792624410299980 T^{6} + 1146403782783 p^{6} T^{8} - 1544490 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
73 \( ( 1 + 1986 T + 1972098 T^{2} + 1660938482 T^{3} + 1247908720527 T^{4} + 781386900229308 T^{5} + 470194412415782204 T^{6} + 781386900229308 p^{3} T^{7} + 1247908720527 p^{6} T^{8} + 1660938482 p^{9} T^{9} + 1972098 p^{12} T^{10} + 1986 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
79 \( ( 1 - 961722 T^{2} + 155618361711 T^{4} + 98219639361604372 T^{6} + 155618361711 p^{6} T^{8} - 961722 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
83 \( 1 - 920473096554 T^{4} + \)\(54\!\cdots\!55\)\( T^{8} - \)\(21\!\cdots\!20\)\( T^{12} + \)\(54\!\cdots\!55\)\( p^{12} T^{16} - 920473096554 p^{24} T^{20} + p^{36} T^{24} \)
89 \( ( 1 + 1873488 T^{2} + 1738766260851 T^{4} + 1272818477897989792 T^{6} + 1738766260851 p^{6} T^{8} + 1873488 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
97 \( ( 1 + 1386 T + 960498 T^{2} + 1082719658 T^{3} - 551766641985 T^{4} - 1656275164022484 T^{5} - 1179485692332035812 T^{6} - 1656275164022484 p^{3} T^{7} - 551766641985 p^{6} T^{8} + 1082719658 p^{9} T^{9} + 960498 p^{12} T^{10} + 1386 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
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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

−3.73472840039101738608881805274, −3.69557882700024288901082343816, −3.59482302884292044209127898460, −3.53840430114179321270488573225, −3.23738335499337420361420216431, −2.98649474856961870713391646610, −2.94861853393238186200285650713, −2.87651144190281946613445622995, −2.84367827431062357289133042853, −2.67227920963814486856359467335, −2.48963329746032895675964270330, −2.30633632096006743832357387893, −2.00419137400384654120926013798, −1.94018277950977876257917637932, −1.74503883455570899848043945379, −1.65952005047355009081023598849, −1.54523838352609036660662393498, −1.47510762155877120049704078552, −1.35795422219092316268719901528, −1.30226608592513718068294178292, −0.855123952330764660102799670898, −0.49017048916585940133722742649, −0.42795333054892905625036077541, −0.13642484230037233851213388650, −0.088076601131234593469996308025, 0.088076601131234593469996308025, 0.13642484230037233851213388650, 0.42795333054892905625036077541, 0.49017048916585940133722742649, 0.855123952330764660102799670898, 1.30226608592513718068294178292, 1.35795422219092316268719901528, 1.47510762155877120049704078552, 1.54523838352609036660662393498, 1.65952005047355009081023598849, 1.74503883455570899848043945379, 1.94018277950977876257917637932, 2.00419137400384654120926013798, 2.30633632096006743832357387893, 2.48963329746032895675964270330, 2.67227920963814486856359467335, 2.84367827431062357289133042853, 2.87651144190281946613445622995, 2.94861853393238186200285650713, 2.98649474856961870713391646610, 3.23738335499337420361420216431, 3.53840430114179321270488573225, 3.59482302884292044209127898460, 3.69557882700024288901082343816, 3.73472840039101738608881805274

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

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