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Properties

Label 24-15e24-1.1-c9e12-0-0
Degree $24$
Conductor $1.683\times 10^{28}$
Sign $1$
Analytic cond. $5.86461\times 10^{24}$
Root an. cond. $10.7648$
Motivic weight $9$
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  + 1.36e3·4-s + 6.26e5·16-s − 1.65e6·19-s − 9.43e6·31-s + 1.62e8·49-s + 7.43e7·61-s + 1.23e8·64-s − 2.25e9·76-s + 9.97e8·79-s + 3.83e9·109-s − 1.04e10·121-s − 1.28e10·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.11e11·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2.66·4-s + 2.39·16-s − 2.90·19-s − 1.83·31-s + 4.02·49-s + 0.687·61-s + 0.922·64-s − 7.74·76-s + 2.88·79-s + 2.60·109-s − 4.45·121-s − 4.88·124-s + 10.4·169-s + 10.7·196-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{24} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(5.86461\times 10^{24}\)
Root analytic conductor: \(10.7648\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{24} \cdot 5^{24} ,\ ( \ : [9/2]^{12} ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(0.001794333428\)
\(L(\frac12)\) \(\approx\) \(0.001794333428\)
\(L(\frac{11}{2})\) 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 \( ( 1 - 341 p T^{2} + 24015 p^{4} T^{4} - 834425 p^{8} T^{6} + 24015 p^{22} T^{8} - 341 p^{37} T^{10} + p^{54} T^{12} )^{2} \)
7 \( ( 1 - 81219767 T^{2} + 109399733453390 p^{2} T^{4} - \)\(10\!\cdots\!35\)\( p^{4} T^{6} + 109399733453390 p^{20} T^{8} - 81219767 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
11 \( ( 1 + 5248652146 T^{2} + 19405248301821496215 T^{4} + \)\(49\!\cdots\!20\)\( T^{6} + 19405248301821496215 p^{18} T^{8} + 5248652146 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
13 \( ( 1 - 55618842363 T^{2} + \)\(13\!\cdots\!10\)\( T^{4} - \)\(18\!\cdots\!15\)\( T^{6} + \)\(13\!\cdots\!10\)\( p^{18} T^{8} - 55618842363 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
17 \( ( 1 - 278868139222 T^{2} + \)\(42\!\cdots\!55\)\( T^{4} - \)\(54\!\cdots\!20\)\( T^{6} + \)\(42\!\cdots\!55\)\( p^{18} T^{8} - 278868139222 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
19 \( ( 1 + 21733 p T + 40996822520 p T^{2} + 260099831988022795 T^{3} + 40996822520 p^{10} T^{4} + 21733 p^{19} T^{5} + p^{27} T^{6} )^{4} \)
23 \( ( 1 - 2559199302138 T^{2} + \)\(14\!\cdots\!55\)\( T^{4} + \)\(32\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!55\)\( p^{18} T^{8} - 2559199302138 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
29 \( ( 1 + 66636150981214 T^{2} + \)\(20\!\cdots\!15\)\( T^{4} + \)\(38\!\cdots\!80\)\( T^{6} + \)\(20\!\cdots\!15\)\( p^{18} T^{8} + 66636150981214 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
31 \( ( 1 + 2357971 T + 42216813067860 T^{2} + \)\(11\!\cdots\!75\)\( T^{3} + 42216813067860 p^{9} T^{4} + 2357971 p^{18} T^{5} + p^{27} T^{6} )^{4} \)
37 \( ( 1 - 549305313046962 T^{2} + \)\(13\!\cdots\!35\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{6} + \)\(13\!\cdots\!35\)\( p^{18} T^{8} - 549305313046962 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
41 \( ( 1 + 1382015265297766 T^{2} + \)\(93\!\cdots\!15\)\( T^{4} + \)\(38\!\cdots\!20\)\( T^{6} + \)\(93\!\cdots\!15\)\( p^{18} T^{8} + 1382015265297766 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
43 \( ( 1 + 986839760363817 T^{2} + \)\(91\!\cdots\!10\)\( T^{4} + \)\(50\!\cdots\!85\)\( T^{6} + \)\(91\!\cdots\!10\)\( p^{18} T^{8} + 986839760363817 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
47 \( ( 1 - 967329592243162 T^{2} - \)\(30\!\cdots\!85\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{6} - \)\(30\!\cdots\!85\)\( p^{18} T^{8} - 967329592243162 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
53 \( ( 1 - 11589443976115438 T^{2} + \)\(60\!\cdots\!15\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{6} + \)\(60\!\cdots\!15\)\( p^{18} T^{8} - 11589443976115438 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
59 \( ( 1 + 28571584252305634 T^{2} + \)\(41\!\cdots\!15\)\( T^{4} + \)\(41\!\cdots\!80\)\( T^{6} + \)\(41\!\cdots\!15\)\( p^{18} T^{8} + 28571584252305634 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
61 \( ( 1 - 18589021 T + 33839980454920970 T^{2} - \)\(42\!\cdots\!65\)\( T^{3} + 33839980454920970 p^{9} T^{4} - 18589021 p^{18} T^{5} + p^{27} T^{6} )^{4} \)
67 \( ( 1 - 152240899529606807 T^{2} + \)\(99\!\cdots\!10\)\( T^{4} - \)\(35\!\cdots\!35\)\( T^{6} + \)\(99\!\cdots\!10\)\( p^{18} T^{8} - 152240899529606807 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
71 \( ( 1 - 46244470732339814 T^{2} + \)\(70\!\cdots\!15\)\( T^{4} - \)\(19\!\cdots\!80\)\( T^{6} + \)\(70\!\cdots\!15\)\( p^{18} T^{8} - 46244470732339814 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
73 \( ( 1 - 209669747046555978 T^{2} + \)\(23\!\cdots\!35\)\( T^{4} - \)\(17\!\cdots\!40\)\( T^{6} + \)\(23\!\cdots\!35\)\( p^{18} T^{8} - 209669747046555978 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
79 \( ( 1 - 249470812 T + 170152625490847005 T^{2} - \)\(77\!\cdots\!20\)\( T^{3} + 170152625490847005 p^{9} T^{4} - 249470812 p^{18} T^{5} + p^{27} T^{6} )^{4} \)
83 \( ( 1 - 366769060122447858 T^{2} + \)\(53\!\cdots\!15\)\( T^{4} - \)\(54\!\cdots\!00\)\( T^{6} + \)\(53\!\cdots\!15\)\( p^{18} T^{8} - 366769060122447858 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
89 \( ( 1 + 103238616270511254 T^{2} + \)\(35\!\cdots\!15\)\( T^{4} + \)\(24\!\cdots\!80\)\( T^{6} + \)\(35\!\cdots\!15\)\( p^{18} T^{8} + 103238616270511254 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
97 \( ( 1 - 1371583538369912427 T^{2} + \)\(16\!\cdots\!10\)\( T^{4} - \)\(11\!\cdots\!35\)\( T^{6} + \)\(16\!\cdots\!10\)\( p^{18} T^{8} - 1371583538369912427 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
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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

−2.78844281965693955512552571163, −2.38600143026294238305321014376, −2.37555779742829433155329143223, −2.02732483006540063180332923246, −2.02496682532906666276365724623, −2.00954399648427942444633155264, −1.97031798931572315993624987698, −1.94762868713476548213468914732, −1.92730239220344579684875241216, −1.92537692398512590481810155681, −1.89171537702490453521957329644, −1.83131537333273525869560620985, −1.46649986541017994780651647301, −1.16421579258466337151929494549, −1.00750240120935459327477704039, −0.981243319831559824586464991625, −0.958586221488613644993907953777, −0.76515677238956964991731106252, −0.74465568547253714428835223862, −0.69560542485090505476738338817, −0.66817243233727739973356854404, −0.54336866745329290565752449815, −0.10682734476247333897714317272, −0.05588464524141480146049024872, −0.00578170334351340284908290258, 0.00578170334351340284908290258, 0.05588464524141480146049024872, 0.10682734476247333897714317272, 0.54336866745329290565752449815, 0.66817243233727739973356854404, 0.69560542485090505476738338817, 0.74465568547253714428835223862, 0.76515677238956964991731106252, 0.958586221488613644993907953777, 0.981243319831559824586464991625, 1.00750240120935459327477704039, 1.16421579258466337151929494549, 1.46649986541017994780651647301, 1.83131537333273525869560620985, 1.89171537702490453521957329644, 1.92537692398512590481810155681, 1.92730239220344579684875241216, 1.94762868713476548213468914732, 1.97031798931572315993624987698, 2.00954399648427942444633155264, 2.02496682532906666276365724623, 2.02732483006540063180332923246, 2.37555779742829433155329143223, 2.38600143026294238305321014376, 2.78844281965693955512552571163

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

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