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Properties

Label 24-65e12-1.1-c1e12-0-0
Degree $24$
Conductor $5.688\times 10^{21}$
Sign $1$
Analytic cond. $0.000382195$
Root an. cond. $0.720435$
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  − 4·4-s − 6·5-s − 6·9-s + 6·16-s + 12·19-s + 24·20-s + 17·25-s + 18·29-s − 16·31-s + 24·36-s + 14·41-s + 36·45-s − 18·49-s − 4·59-s + 6·61-s + 6·64-s − 12·71-s − 48·76-s − 104·79-s − 36·80-s + 26·81-s + 20·89-s − 72·95-s − 68·100-s − 26·101-s + 24·109-s − 72·116-s + ⋯
L(s)  = 1  − 2·4-s − 2.68·5-s − 2·9-s + 3/2·16-s + 2.75·19-s + 5.36·20-s + 17/5·25-s + 3.34·29-s − 2.87·31-s + 4·36-s + 2.18·41-s + 5.36·45-s − 2.57·49-s − 0.520·59-s + 0.768·61-s + 3/4·64-s − 1.42·71-s − 5.50·76-s − 11.7·79-s − 4.02·80-s + 26/9·81-s + 2.11·89-s − 7.38·95-s − 6.79·100-s − 2.58·101-s + 2.29·109-s − 6.68·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(5^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(0.000382195\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 5^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.08870521842\)
\(L(\frac12)\) \(\approx\) \(0.08870521842\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + 3 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 15 T^{2} + 3 p T^{4} - 322 T^{6} + 3 p^{3} T^{8} - 15 p^{4} T^{10} + p^{6} T^{12} \)
good2 \( 1 + p^{2} T^{2} + 5 p T^{4} + 5 p T^{6} - p^{4} T^{8} - 43 p T^{10} - 223 T^{12} - 43 p^{3} T^{14} - p^{8} T^{16} + 5 p^{7} T^{18} + 5 p^{9} T^{20} + p^{12} T^{22} + p^{12} T^{24} \)
3 \( 1 + 2 p T^{2} + 10 T^{4} - 40 T^{6} - 146 T^{8} + 226 T^{10} + 2062 T^{12} + 226 p^{2} T^{14} - 146 p^{4} T^{16} - 40 p^{6} T^{18} + 10 p^{8} T^{20} + 2 p^{11} T^{22} + p^{12} T^{24} \)
7 \( 1 + 18 T^{2} + 82 T^{4} + 536 T^{6} + 12326 T^{8} + 78550 T^{10} + 235838 T^{12} + 78550 p^{2} T^{14} + 12326 p^{4} T^{16} + 536 p^{6} T^{18} + 82 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24} \)
11 \( ( 1 - 20 T^{2} - 16 T^{3} + 180 T^{4} + 160 T^{5} - 1674 T^{6} + 160 p T^{7} + 180 p^{2} T^{8} - 16 p^{3} T^{9} - 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 + 67 T^{2} + 2371 T^{4} + 56020 T^{6} + 996641 T^{8} + 14527513 T^{10} + 221018582 T^{12} + 14527513 p^{2} T^{14} + 996641 p^{4} T^{16} + 56020 p^{6} T^{18} + 2371 p^{8} T^{20} + 67 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 - 6 T - 20 T^{2} + 100 T^{3} + 764 T^{4} - 1606 T^{5} - 11338 T^{6} - 1606 p T^{7} + 764 p^{2} T^{8} + 100 p^{3} T^{9} - 20 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 126 T^{2} + 9010 T^{4} + 451400 T^{6} + 17445734 T^{8} + 540254266 T^{10} + 13705453502 T^{12} + 540254266 p^{2} T^{14} + 17445734 p^{4} T^{16} + 451400 p^{6} T^{18} + 9010 p^{8} T^{20} + 126 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{6} \)
31 \( ( 1 + 4 T + 53 T^{2} + 288 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
37 \( 1 + 187 T^{2} + 19451 T^{4} + 1426940 T^{6} + 81658601 T^{8} + 3849109833 T^{10} + 153778997622 T^{12} + 3849109833 p^{2} T^{14} + 81658601 p^{4} T^{16} + 1426940 p^{6} T^{18} + 19451 p^{8} T^{20} + 187 p^{10} T^{22} + p^{12} T^{24} \)
41 \( ( 1 - 7 T - 45 T^{2} + 500 T^{3} + 929 T^{4} - 12237 T^{5} + 28438 T^{6} - 12237 p T^{7} + 929 p^{2} T^{8} + 500 p^{3} T^{9} - 45 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
43 \( 1 + 178 T^{2} + 17426 T^{4} + 1084520 T^{6} + 47574566 T^{8} + 1584620742 T^{10} + 56532979182 T^{12} + 1584620742 p^{2} T^{14} + 47574566 p^{4} T^{16} + 1084520 p^{6} T^{18} + 17426 p^{8} T^{20} + 178 p^{10} T^{22} + p^{12} T^{24} \)
47 \( ( 1 - 46 T^{2} + 3407 T^{4} - 54276 T^{6} + 3407 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 147 T^{2} + 6923 T^{4} - 205346 T^{6} + 6923 p^{2} T^{8} - 147 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 + 2 T - 2 p T^{2} + 44 T^{3} + 7486 T^{4} - 9630 T^{5} - 471322 T^{6} - 9630 p T^{7} + 7486 p^{2} T^{8} + 44 p^{3} T^{9} - 2 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 3 T - 125 T^{2} + 100 T^{3} + 9029 T^{4} + 4247 T^{5} - 611842 T^{6} + 4247 p T^{7} + 9029 p^{2} T^{8} + 100 p^{3} T^{9} - 125 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 302 T^{2} + 48066 T^{4} + 5726680 T^{6} + 564790166 T^{8} + 46598346138 T^{10} + 3315667632142 T^{12} + 46598346138 p^{2} T^{14} + 564790166 p^{4} T^{16} + 5726680 p^{6} T^{18} + 48066 p^{8} T^{20} + 302 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 + 6 T - 176 T^{2} - 380 T^{3} + 24936 T^{4} + 24734 T^{5} - 1974402 T^{6} + 24734 p T^{7} + 24936 p^{2} T^{8} - 380 p^{3} T^{9} - 176 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 223 T^{2} + 31055 T^{4} - 2685330 T^{6} + 31055 p^{2} T^{8} - 223 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 + 26 T + 417 T^{2} + 4268 T^{3} + 417 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( ( 1 - 222 T^{2} + 35303 T^{4} - 3305156 T^{6} + 35303 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 10 T - 10 T^{2} - 8 p T^{3} + 370 T^{4} + 95250 T^{5} - 421426 T^{6} + 95250 p T^{7} + 370 p^{2} T^{8} - 8 p^{4} T^{9} - 10 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( 1 + 302 T^{2} + 44346 T^{4} + 4094200 T^{6} + 2434958 p T^{8} - 3936373062 T^{10} - 1781970067778 T^{12} - 3936373062 p^{2} T^{14} + 2434958 p^{5} T^{16} + 4094200 p^{6} T^{18} + 44346 p^{8} T^{20} + 302 p^{10} T^{22} + p^{12} T^{24} \)
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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

−5.75786958389092996629339701671, −5.43924391818933259494223512527, −5.36259536706706809649376147535, −5.28749380845141342223769014597, −5.28589253151984631524995208453, −4.90208496999044460051823758533, −4.73381412011506811017839262963, −4.57110447163098693915333919774, −4.53893223954281904920732777125, −4.46893513135101342848555581271, −4.36765884739908055310249080766, −4.30109500823462424454792714020, −3.92706211960688257891627126189, −3.84729037660810411257783261096, −3.76997695851957584202444893920, −3.32363031146889635095663936151, −3.23694847152146753588120885048, −3.14663487049026575016803075665, −3.08511671997933752977749102447, −2.88873652432080843097933219345, −2.88781723689670296305944389893, −2.46051051223476293834244925197, −1.82055882492538526323385586890, −1.71459040122033398318564720035, −0.833996455731397483900533023978, 0.833996455731397483900533023978, 1.71459040122033398318564720035, 1.82055882492538526323385586890, 2.46051051223476293834244925197, 2.88781723689670296305944389893, 2.88873652432080843097933219345, 3.08511671997933752977749102447, 3.14663487049026575016803075665, 3.23694847152146753588120885048, 3.32363031146889635095663936151, 3.76997695851957584202444893920, 3.84729037660810411257783261096, 3.92706211960688257891627126189, 4.30109500823462424454792714020, 4.36765884739908055310249080766, 4.46893513135101342848555581271, 4.53893223954281904920732777125, 4.57110447163098693915333919774, 4.73381412011506811017839262963, 4.90208496999044460051823758533, 5.28589253151984631524995208453, 5.28749380845141342223769014597, 5.36259536706706809649376147535, 5.43924391818933259494223512527, 5.75786958389092996629339701671

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

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