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Properties

Label 24-60e12-1.1-c1e12-0-0
Degree $24$
Conductor $2.177\times 10^{21}$
Sign $1$
Analytic cond. $0.000146264$
Root an. cond. $0.692172$
Motivic weight $1$
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  − 4·8-s − 4·13-s + 3·16-s − 20·17-s − 10·25-s + 4·32-s + 4·37-s + 16·41-s + 4·53-s − 32·61-s + 8·64-s + 44·73-s − 3·81-s − 20·97-s − 40·101-s + 16·104-s − 52·113-s + 60·121-s − 16·125-s + 127-s − 16·128-s + 131-s + 80·136-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.41·8-s − 1.10·13-s + 3/4·16-s − 4.85·17-s − 2·25-s + 0.707·32-s + 0.657·37-s + 2.49·41-s + 0.549·53-s − 4.09·61-s + 64-s + 5.14·73-s − 1/3·81-s − 2.03·97-s − 3.98·101-s + 1.56·104-s − 4.89·113-s + 5.45·121-s − 1.43·125-s + 0.0887·127-s − 1.41·128-s + 0.0873·131-s + 6.85·136-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(0.000146264\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{12} \cdot 5^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1709821360\)
\(L(\frac12)\) \(\approx\) \(0.1709821360\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{2} T^{3} - 3 T^{4} - p^{2} T^{5} + p^{3} T^{6} - p^{3} T^{7} - 3 p^{2} T^{8} + p^{5} T^{9} + p^{6} T^{12} \)
3 \( ( 1 + T^{4} )^{3} \)
5 \( ( 1 + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} )^{2} \)
good7 \( 1 - 66 T^{4} + 671 T^{8} + 41476 T^{12} + 671 p^{4} T^{16} - 66 p^{8} T^{20} + p^{12} T^{24} \)
11 \( ( 1 - 30 T^{2} + 491 T^{4} - 6076 T^{6} + 491 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
13 \( ( 1 + 2 T + 2 T^{2} - 6 T^{3} + 27 T^{4} + 580 T^{5} + 1124 T^{6} + 580 p T^{7} + 27 p^{2} T^{8} - 6 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
17 \( ( 1 + 10 T + 50 T^{2} + 250 T^{3} + 1155 T^{4} + 4260 T^{5} + 16100 T^{6} + 4260 p T^{7} + 1155 p^{2} T^{8} + 250 p^{3} T^{9} + 50 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
19 \( ( 1 + 74 T^{2} + 2775 T^{4} + 65228 T^{6} + 2775 p^{2} T^{8} + 74 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
23 \( 1 + 1190 T^{4} + 607727 T^{8} + 223349076 T^{12} + 607727 p^{4} T^{16} + 1190 p^{8} T^{20} + p^{12} T^{24} \)
29 \( ( 1 - 154 T^{2} + 10395 T^{4} - 392596 T^{6} + 10395 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - 74 T^{2} + 4175 T^{4} - 143436 T^{6} + 4175 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 10 T + 93 T^{2} - 472 T^{3} + 93 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2}( 1 + 8 T - 11 T^{2} - 424 T^{3} - 11 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( ( 1 - 4 T + 103 T^{2} - 264 T^{3} + 103 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
43 \( 1 - 6826 T^{4} + 23542879 T^{8} - 52748850124 T^{12} + 23542879 p^{4} T^{16} - 6826 p^{8} T^{20} + p^{12} T^{24} \)
47 \( 1 + 1606 T^{4} + 1555663 T^{8} - 11534742380 T^{12} + 1555663 p^{4} T^{16} + 1606 p^{8} T^{20} + p^{12} T^{24} \)
53 \( ( 1 - 2 T + 2 T^{2} - 90 T^{3} + 5291 T^{4} - 7252 T^{5} + 7972 T^{6} - 7252 p T^{7} + 5291 p^{2} T^{8} - 90 p^{3} T^{9} + 2 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
59 \( ( 1 + 254 T^{2} + 30475 T^{4} + 2237948 T^{6} + 30475 p^{2} T^{8} + 254 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 8 T + 83 T^{2} + 1152 T^{3} + 83 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
67 \( 1 - 6346 T^{4} + 60212671 T^{8} - 234156159244 T^{12} + 60212671 p^{4} T^{16} - 6346 p^{8} T^{20} + p^{12} T^{24} \)
71 \( ( 1 - 170 T^{2} + 18527 T^{4} - 1427916 T^{6} + 18527 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 22 T + 242 T^{2} - 2246 T^{3} + 20271 T^{4} - 179604 T^{5} + 1567964 T^{6} - 179604 p T^{7} + 20271 p^{2} T^{8} - 2246 p^{3} T^{9} + 242 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 + 170 T^{2} + 10415 T^{4} + 507660 T^{6} + 10415 p^{2} T^{8} + 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( 1 + 7542 T^{4} - 16589569 T^{8} - 295904039564 T^{12} - 16589569 p^{4} T^{16} + 7542 p^{8} T^{20} + p^{12} T^{24} \)
89 \( ( 1 - 462 T^{2} + 94223 T^{4} - 10861604 T^{6} + 94223 p^{2} T^{8} - 462 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( ( 1 + 10 T + 50 T^{2} - 1078 T^{3} - 9153 T^{4} + 45804 T^{5} + 1496732 T^{6} + 45804 p T^{7} - 9153 p^{2} T^{8} - 1078 p^{3} T^{9} + 50 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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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

−5.80193236186153991877192141527, −5.64597234276793293522754012365, −5.57950822027377763077666186519, −5.43689528568042674304442276403, −5.35152973487716363715185917574, −5.12045987168770241831574206311, −5.00239101024269964312071468942, −4.86935030706652868948885848305, −4.53011808042202940201052835960, −4.42259064662844709064730415666, −4.35002774704167954760368500884, −4.22186681257837288851574594309, −4.10845597245463350886600356084, −4.10820850581918588614915726140, −3.96505536266266673223687962790, −3.38477227989812797661300606691, −3.37130257409424508583511936349, −3.23476554326180866955618992189, −2.99794960363558667981740078565, −2.62726573415258421154532584130, −2.45183576347053094532963864216, −2.42227100475349896467561560502, −2.35091786896788376344780558035, −1.87502238686416301347343536366, −1.63912336428237542663196072666, 1.63912336428237542663196072666, 1.87502238686416301347343536366, 2.35091786896788376344780558035, 2.42227100475349896467561560502, 2.45183576347053094532963864216, 2.62726573415258421154532584130, 2.99794960363558667981740078565, 3.23476554326180866955618992189, 3.37130257409424508583511936349, 3.38477227989812797661300606691, 3.96505536266266673223687962790, 4.10820850581918588614915726140, 4.10845597245463350886600356084, 4.22186681257837288851574594309, 4.35002774704167954760368500884, 4.42259064662844709064730415666, 4.53011808042202940201052835960, 4.86935030706652868948885848305, 5.00239101024269964312071468942, 5.12045987168770241831574206311, 5.35152973487716363715185917574, 5.43689528568042674304442276403, 5.57950822027377763077666186519, 5.64597234276793293522754012365, 5.80193236186153991877192141527

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

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