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Properties

Label 24-252e12-1.1-c10e12-0-1
Degree $24$
Conductor $6.559\times 10^{28}$
Sign $1$
Analytic cond. $2.83806\times 10^{26}$
Root an. cond. $12.6534$
Motivic weight $10$
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  + 2.57e4·7-s + 2.99e7·25-s + 1.79e8·37-s − 1.04e9·43-s + 7.55e8·49-s + 8.00e9·67-s − 1.01e10·79-s − 7.73e10·109-s − 1.16e11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.83e11·169-s + 173-s + 7.73e11·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 1.53·7-s + 3.07·25-s + 2.59·37-s − 7.07·43-s + 2.67·49-s + 5.92·67-s − 3.28·79-s − 5.02·109-s − 4.48·121-s + 3.50·169-s + 4.71·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{24} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(2.83806\times 10^{26}\)
Root analytic conductor: \(12.6534\)
Motivic weight: \(10\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [5]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(5.919030330\)
\(L(\frac12)\) \(\approx\) \(5.919030330\)
\(L(6)\) 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 \( 1 \)
7 \( ( 1 - 1842 p T - 53421 p^{4} T^{2} + 525956 p^{8} T^{3} - 53421 p^{14} T^{4} - 1842 p^{21} T^{5} + p^{30} T^{6} )^{2} \)
good5 \( ( 1 - 2998458 p T^{2} + 10340989803243 p^{2} T^{4} - 22588232794850106196 p^{3} T^{6} + 10340989803243 p^{22} T^{8} - 2998458 p^{41} T^{10} + p^{60} T^{12} )^{2} \)
11 \( ( 1 + 58223676066 T^{2} + \)\(24\!\cdots\!55\)\( T^{4} + \)\(71\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!55\)\( p^{20} T^{8} + 58223676066 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
13 \( ( 1 - 241613202198 T^{2} + \)\(13\!\cdots\!71\)\( T^{4} + \)\(62\!\cdots\!48\)\( T^{6} + \)\(13\!\cdots\!71\)\( p^{20} T^{8} - 241613202198 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
17 \( ( 1 - 356486261202 p T^{2} + \)\(19\!\cdots\!35\)\( T^{4} - \)\(43\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!35\)\( p^{20} T^{8} - 356486261202 p^{41} T^{10} + p^{60} T^{12} )^{2} \)
19 \( ( 1 - 24097062089262 T^{2} + \)\(29\!\cdots\!51\)\( T^{4} - \)\(62\!\cdots\!68\)\( p^{2} T^{6} + \)\(29\!\cdots\!51\)\( p^{20} T^{8} - 24097062089262 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
23 \( ( 1 + 48424030213074 T^{2} + \)\(36\!\cdots\!35\)\( T^{4} + \)\(64\!\cdots\!40\)\( p T^{6} + \)\(36\!\cdots\!35\)\( p^{20} T^{8} + 48424030213074 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
29 \( ( 1 + 652961297646006 T^{2} + \)\(41\!\cdots\!55\)\( T^{4} + \)\(13\!\cdots\!40\)\( T^{6} + \)\(41\!\cdots\!55\)\( p^{20} T^{8} + 652961297646006 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
31 \( ( 1 - 795610800245382 T^{2} - \)\(45\!\cdots\!29\)\( T^{4} + \)\(99\!\cdots\!92\)\( T^{6} - \)\(45\!\cdots\!29\)\( p^{20} T^{8} - 795610800245382 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
37 \( ( 1 - 44981382 T + 3300515410775595 T^{2} + \)\(49\!\cdots\!40\)\( T^{3} + 3300515410775595 p^{10} T^{4} - 44981382 p^{20} T^{5} + p^{30} T^{6} )^{4} \)
41 \( ( 1 - 40016540963190546 T^{2} + \)\(99\!\cdots\!75\)\( T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(99\!\cdots\!75\)\( p^{20} T^{8} - 40016540963190546 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
43 \( ( 1 + 260181954 T + 73032022476603819 T^{2} + \)\(11\!\cdots\!24\)\( T^{3} + 73032022476603819 p^{10} T^{4} + 260181954 p^{20} T^{5} + p^{30} T^{6} )^{4} \)
47 \( ( 1 - 156520773272742534 T^{2} + \)\(14\!\cdots\!95\)\( T^{4} - \)\(93\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!95\)\( p^{20} T^{8} - 156520773272742534 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
53 \( ( 1 - 20385857014137066 T^{2} + \)\(87\!\cdots\!35\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{6} + \)\(87\!\cdots\!35\)\( p^{20} T^{8} - 20385857014137066 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
59 \( ( 1 - 2279081470229631126 T^{2} + \)\(24\!\cdots\!75\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(24\!\cdots\!75\)\( p^{20} T^{8} - 2279081470229631126 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
61 \( ( 1 - 3556138438413005142 T^{2} + \)\(57\!\cdots\!11\)\( T^{4} - \)\(52\!\cdots\!88\)\( T^{6} + \)\(57\!\cdots\!11\)\( p^{20} T^{8} - 3556138438413005142 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
67 \( ( 1 - 2001128466 T + 4909178112181776219 T^{2} - \)\(56\!\cdots\!96\)\( T^{3} + 4909178112181776219 p^{10} T^{4} - 2001128466 p^{20} T^{5} + p^{30} T^{6} )^{4} \)
71 \( ( 1 + 8615030182813304466 T^{2} + \)\(41\!\cdots\!75\)\( T^{4} + \)\(14\!\cdots\!60\)\( T^{6} + \)\(41\!\cdots\!75\)\( p^{20} T^{8} + 8615030182813304466 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
73 \( ( 1 - 16184053668127116774 T^{2} + \)\(11\!\cdots\!95\)\( T^{4} - \)\(54\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!95\)\( p^{20} T^{8} - 16184053668127116774 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
79 \( ( 1 + 2527045806 T + 28450128328238096835 T^{2} + \)\(45\!\cdots\!80\)\( T^{3} + 28450128328238096835 p^{10} T^{4} + 2527045806 p^{20} T^{5} + p^{30} T^{6} )^{4} \)
83 \( ( 1 - 6549233523129132534 T^{2} + \)\(15\!\cdots\!55\)\( T^{4} - \)\(59\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!55\)\( p^{20} T^{8} - 6549233523129132534 p^{40} T^{10} + p^{60} T^{12} )^{2} \)
89 \( ( 1 - \)\(11\!\cdots\!06\)\( T^{2} + \)\(65\!\cdots\!55\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(65\!\cdots\!55\)\( p^{20} T^{8} - \)\(11\!\cdots\!06\)\( p^{40} T^{10} + p^{60} T^{12} )^{2} \)
97 \( ( 1 - \)\(31\!\cdots\!58\)\( T^{2} + \)\(46\!\cdots\!51\)\( T^{4} - \)\(41\!\cdots\!92\)\( T^{6} + \)\(46\!\cdots\!51\)\( p^{20} T^{8} - \)\(31\!\cdots\!58\)\( p^{40} T^{10} + p^{60} T^{12} )^{2} \)
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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

−2.55070179574202951765300484657, −2.32765302162443461146184070315, −2.22357456072323612047498583403, −2.14891847217629430684355258144, −2.06127848884698900582385727200, −2.02328820805781800747156282758, −2.00097495243258225597681873580, −1.90359628726526107716650291475, −1.54150101731786029223146793000, −1.39812860969263087218124411977, −1.37033981774694273231632289305, −1.34392664001085816329901846690, −1.33901854010826807070909472675, −1.32255646981855111549453935268, −1.17637119983094039542363549755, −1.02347900241143503843450366168, −0.962907423945166156070890181914, −0.825172568639039704014984517220, −0.60508218527608744497868433518, −0.50036846618815776654387538250, −0.48058641070650858847246207090, −0.40330455045311530366357283607, −0.35661842410798682869087967772, −0.15081335975508964194740959642, −0.05933655218717483759763188733, 0.05933655218717483759763188733, 0.15081335975508964194740959642, 0.35661842410798682869087967772, 0.40330455045311530366357283607, 0.48058641070650858847246207090, 0.50036846618815776654387538250, 0.60508218527608744497868433518, 0.825172568639039704014984517220, 0.962907423945166156070890181914, 1.02347900241143503843450366168, 1.17637119983094039542363549755, 1.32255646981855111549453935268, 1.33901854010826807070909472675, 1.34392664001085816329901846690, 1.37033981774694273231632289305, 1.39812860969263087218124411977, 1.54150101731786029223146793000, 1.90359628726526107716650291475, 2.00097495243258225597681873580, 2.02328820805781800747156282758, 2.06127848884698900582385727200, 2.14891847217629430684355258144, 2.22357456072323612047498583403, 2.32765302162443461146184070315, 2.55070179574202951765300484657

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

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