Properties

Label 24-735e12-1.1-c1e12-0-1
Degree $24$
Conductor $2.486\times 10^{34}$
Sign $1$
Analytic cond. $1.67021\times 10^{9}$
Root an. cond. $2.42260$
Motivic weight $1$
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  − 4-s + 2·5-s + 3·9-s − 12·11-s − 16-s − 12·19-s − 2·20-s + 3·25-s − 8·29-s + 4·31-s − 3·36-s − 8·41-s + 12·44-s + 6·45-s − 24·55-s − 32·59-s − 12·61-s + 10·64-s + 24·71-s + 12·76-s + 24·79-s − 2·80-s + 3·81-s − 28·89-s − 24·95-s − 36·99-s − 3·100-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 9-s − 3.61·11-s − 1/4·16-s − 2.75·19-s − 0.447·20-s + 3/5·25-s − 1.48·29-s + 0.718·31-s − 1/2·36-s − 1.24·41-s + 1.80·44-s + 0.894·45-s − 3.23·55-s − 4.16·59-s − 1.53·61-s + 5/4·64-s + 2.84·71-s + 1.37·76-s + 2.70·79-s − 0.223·80-s + 1/3·81-s − 2.96·89-s − 2.46·95-s − 3.61·99-s − 0.299·100-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 5^{12} \cdot 7^{24}\)
Sign: $1$
Analytic conductor: \(1.67021\times 10^{9}\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 5^{12} \cdot 7^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.582520569\)
\(L(\frac12)\) \(\approx\) \(1.582520569\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T^{2} + T^{4} )^{3} \)
5 \( 1 - 2 T + T^{2} + 18 T^{3} - 6 p T^{4} - 26 T^{5} + 249 T^{6} - 26 p T^{7} - 6 p^{3} T^{8} + 18 p^{3} T^{9} + p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 \)
good2 \( 1 + T^{2} + p T^{4} - 7 T^{6} + p T^{8} + 21 T^{10} + 117 T^{12} + 21 p^{2} T^{14} + p^{5} T^{16} - 7 p^{6} T^{18} + p^{9} T^{20} + p^{10} T^{22} + p^{12} T^{24} \)
11 \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{6} \)
13 \( ( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 359 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 + 70 T^{2} + 2485 T^{4} + 66610 T^{6} + 1548890 T^{8} + 31831870 T^{10} + 576997613 T^{12} + 31831870 p^{2} T^{14} + 1548890 p^{4} T^{16} + 66610 p^{6} T^{18} + 2485 p^{8} T^{20} + 70 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 + 6 T - 17 T^{2} - 58 T^{3} + 674 T^{4} + 46 T^{5} - 17305 T^{6} + 46 p T^{7} + 674 p^{2} T^{8} - 58 p^{3} T^{9} - 17 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 106 T^{2} + 6053 T^{4} + 248702 T^{6} + 8184794 T^{8} + 227653746 T^{10} + 5545513725 T^{12} + 227653746 p^{2} T^{14} + 8184794 p^{4} T^{16} + 248702 p^{6} T^{18} + 6053 p^{8} T^{20} + 106 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
31 \( ( 1 - 2 T - 37 T^{2} - 202 T^{3} + 530 T^{4} + 5622 T^{5} + 7227 T^{6} + 5622 p T^{7} + 530 p^{2} T^{8} - 202 p^{3} T^{9} - 37 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( 1 + 46 T^{2} + 717 T^{4} - 85222 T^{6} - 3398278 T^{8} + 14638566 T^{10} + 4079564917 T^{12} + 14638566 p^{2} T^{14} - 3398278 p^{4} T^{16} - 85222 p^{6} T^{18} + 717 p^{8} T^{20} + 46 p^{10} T^{22} + p^{12} T^{24} \)
41 \( ( 1 + 2 T + 63 T^{2} - 36 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
43 \( ( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 2839 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( 1 + 154 T^{2} + 11573 T^{4} + 565358 T^{6} + 20169434 T^{8} + 410981154 T^{10} + 5646494445 T^{12} + 410981154 p^{2} T^{14} + 20169434 p^{4} T^{16} + 565358 p^{6} T^{18} + 11573 p^{8} T^{20} + 154 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 146 T^{2} + 7213 T^{4} + 289382 T^{6} + 30094394 T^{8} + 2063064026 T^{10} + 100360862165 T^{12} + 2063064026 p^{2} T^{14} + 30094394 p^{4} T^{16} + 289382 p^{6} T^{18} + 7213 p^{8} T^{20} + 146 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 + 16 T + 143 T^{2} + 592 T^{3} - 3626 T^{4} - 88944 T^{5} - 864085 T^{6} - 88944 p T^{7} - 3626 p^{2} T^{8} + 592 p^{3} T^{9} + 143 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 6 T - 95 T^{2} - 182 T^{3} + 6266 T^{4} - 10162 T^{5} - 492559 T^{6} - 10162 p T^{7} + 6266 p^{2} T^{8} - 182 p^{3} T^{9} - 95 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 274 T^{2} + 38973 T^{4} + 3966278 T^{6} + 329505914 T^{8} + 23681216154 T^{10} + 1596172229125 T^{12} + 23681216154 p^{2} T^{14} + 329505914 p^{4} T^{16} + 3966278 p^{6} T^{18} + 38973 p^{8} T^{20} + 274 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 - 2 T + p T^{2} )^{12} \)
73 \( 1 + 298 T^{2} + 45029 T^{4} + 5080382 T^{6} + 496253018 T^{8} + 43754080818 T^{10} + 3427677334845 T^{12} + 43754080818 p^{2} T^{14} + 496253018 p^{4} T^{16} + 5080382 p^{6} T^{18} + 45029 p^{8} T^{20} + 298 p^{10} T^{22} + p^{12} T^{24} \)
79 \( ( 1 - 12 T - 77 T^{2} + 500 T^{3} + 12470 T^{4} - 4172 T^{5} - 1287289 T^{6} - 4172 p T^{7} + 12470 p^{2} T^{8} + 500 p^{3} T^{9} - 77 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( ( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 + 14 T - 123 T^{2} - 598 T^{3} + 37922 T^{4} + 123654 T^{5} - 2788283 T^{6} + 123654 p T^{7} + 37922 p^{2} T^{8} - 598 p^{3} T^{9} - 123 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( ( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 8719 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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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

−3.31682875215669431290018080702, −3.15490486241868750903995000822, −3.14038007154218652060181379474, −3.04264921476603153708907347099, −2.98584515500562604811386691058, −2.95034239057800090968795802966, −2.81925851733273917690794422459, −2.59033130072563930482331407928, −2.53904077060630522528546973202, −2.35874467084664345462365966866, −2.27186352527307599287460259241, −2.22375178452815993139127225479, −2.03633243805403676406008683384, −1.93438516303377575509849363484, −1.92401266069511684911111986296, −1.86675585616658081983532940485, −1.72446712031515504860532106209, −1.46216831718577683317953241346, −1.40025572456451721798023501384, −1.21892271869772951239904762424, −1.05155817033895818498382416126, −0.58790996674027425526039063301, −0.47854065465184582465906964037, −0.46067974536858599037851040752, −0.18362033935597325074721898590, 0.18362033935597325074721898590, 0.46067974536858599037851040752, 0.47854065465184582465906964037, 0.58790996674027425526039063301, 1.05155817033895818498382416126, 1.21892271869772951239904762424, 1.40025572456451721798023501384, 1.46216831718577683317953241346, 1.72446712031515504860532106209, 1.86675585616658081983532940485, 1.92401266069511684911111986296, 1.93438516303377575509849363484, 2.03633243805403676406008683384, 2.22375178452815993139127225479, 2.27186352527307599287460259241, 2.35874467084664345462365966866, 2.53904077060630522528546973202, 2.59033130072563930482331407928, 2.81925851733273917690794422459, 2.95034239057800090968795802966, 2.98584515500562604811386691058, 3.04264921476603153708907347099, 3.14038007154218652060181379474, 3.15490486241868750903995000822, 3.31682875215669431290018080702

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

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