Properties

Label 16-1050e8-1.1-c2e8-0-3
Degree $16$
Conductor $1.477\times 10^{24}$
Sign $1$
Analytic cond. $4.48946\times 10^{11}$
Root an. cond. $5.34887$
Motivic weight $2$
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 − 6·9-s − 24·11-s + 4·16-s + 84·19-s + 204·31-s − 24·36-s − 96·44-s + 2·49-s + 48·59-s − 144·61-s − 16·64-s + 624·71-s + 336·76-s + 20·79-s + 9·81-s + 144·89-s + 144·99-s + 72·101-s + 580·109-s + 700·121-s + 816·124-s + 127-s + 131-s + 137-s + 139-s − 24·144-s + ⋯
L(s)  = 1  + 4-s − 2/3·9-s − 2.18·11-s + 1/4·16-s + 4.42·19-s + 6.58·31-s − 2/3·36-s − 2.18·44-s + 2/49·49-s + 0.813·59-s − 2.36·61-s − 1/4·64-s + 8.78·71-s + 4.42·76-s + 0.253·79-s + 1/9·81-s + 1.61·89-s + 1.45·99-s + 0.712·101-s + 5.32·109-s + 5.78·121-s + 6.58·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1/6·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(4.48946\times 10^{11}\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1050} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.865607681\)
\(L(\frac12)\) \(\approx\) \(2.865607681\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
3 \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
5 \( 1 \)
7 \( 1 - 2 T^{2} - 2397 T^{4} - 2 p^{4} T^{6} + p^{8} T^{8} \)
good11 \( ( 1 + 6 T - 85 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
13 \( ( 1 - 98 T^{2} + 54915 T^{4} - 98 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( 1 - 724 T^{2} + 244522 T^{4} - 81531088 T^{6} + 27450898579 T^{8} - 81531088 p^{4} T^{10} + 244522 p^{8} T^{12} - 724 p^{12} T^{14} + p^{16} T^{16} \)
19 \( ( 1 - 42 T + 1433 T^{2} - 35490 T^{3} + 795972 T^{4} - 35490 p^{2} T^{5} + 1433 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 + 532 T^{2} + 25834 T^{4} - 160925744 T^{6} - 91366008365 T^{8} - 160925744 p^{4} T^{10} + 25834 p^{8} T^{12} + 532 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{4} \)
31 \( ( 1 - 102 T + 6041 T^{2} - 8466 p T^{3} + 9396 p^{2} T^{4} - 8466 p^{3} T^{5} + 6041 p^{4} T^{6} - 102 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( 1 + 4658 T^{2} + 12663793 T^{4} + 24616826642 T^{6} + 37422095114116 T^{8} + 24616826642 p^{4} T^{10} + 12663793 p^{8} T^{12} + 4658 p^{12} T^{14} + p^{16} T^{16} \)
41 \( ( 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 - 7154 T^{2} + 19618419 T^{4} - 7154 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( 1 - 5884 T^{2} + 16346122 T^{4} - 50107979248 T^{6} + 141234885114259 T^{8} - 50107979248 p^{4} T^{10} + 16346122 p^{8} T^{12} - 5884 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 + 2740 T^{2} - 15454 p T^{4} - 20424782000 T^{6} - 52164965504717 T^{8} - 20424782000 p^{4} T^{10} - 15454 p^{9} T^{12} + 2740 p^{12} T^{14} + p^{16} T^{16} \)
59 \( ( 1 - 24 T + 6026 T^{2} - 140016 T^{3} + 22586547 T^{4} - 140016 p^{2} T^{5} + 6026 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
61 \( ( 1 + 72 T + 9218 T^{2} + 539280 T^{3} + 48684147 T^{4} + 539280 p^{2} T^{5} + 9218 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 + 4850 T^{2} + 20028433 T^{4} - 178519648750 T^{6} - 867863917078652 T^{8} - 178519648750 p^{4} T^{10} + 20028433 p^{8} T^{12} + 4850 p^{12} T^{14} + p^{16} T^{16} \)
71 \( ( 1 - 156 T + 178 p T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
73 \( 1 - 1390 T^{2} - 41408207 T^{4} + 18704083250 T^{6} + 1017053877524068 T^{8} + 18704083250 p^{4} T^{10} - 41408207 p^{8} T^{12} - 1390 p^{12} T^{14} + p^{16} T^{16} \)
79 \( ( 1 - 10 T - 3695 T^{2} + 86870 T^{3} - 25172156 T^{4} + 86870 p^{2} T^{5} - 3695 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( ( 1 + 116 p T^{2} + 107694438 T^{4} + 116 p^{5} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 36 T + 8353 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
97 \( ( 1 + 36580 T^{2} + 511416774 T^{4} + 36580 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.93483995836902272256927728580, −3.75033818031506282456090147919, −3.72282298382036659245105463868, −3.40396672457536603883092739752, −3.38627102339797634936611315838, −3.30410531069305064906965322222, −3.17462537696018144300319306397, −3.03466553655436401605943432713, −3.02676227075413473795041790642, −2.89600400218659627488735502906, −2.59614708834694592103916590168, −2.42862841591821593245333338417, −2.38634561439140460998650701089, −2.36285830344715332842668071881, −2.05255201060356710295083610828, −1.97893248775238046285424303892, −1.83218759756653993708428166508, −1.66149292151577714623247521555, −1.02784114524428917915344318758, −1.01139455512015196930870956304, −0.966127967918405371616556982602, −0.78343264446373099636293048048, −0.71830839443367901878976019797, −0.71331569752112497765568035548, −0.07250510477860099331002743596, 0.07250510477860099331002743596, 0.71331569752112497765568035548, 0.71830839443367901878976019797, 0.78343264446373099636293048048, 0.966127967918405371616556982602, 1.01139455512015196930870956304, 1.02784114524428917915344318758, 1.66149292151577714623247521555, 1.83218759756653993708428166508, 1.97893248775238046285424303892, 2.05255201060356710295083610828, 2.36285830344715332842668071881, 2.38634561439140460998650701089, 2.42862841591821593245333338417, 2.59614708834694592103916590168, 2.89600400218659627488735502906, 3.02676227075413473795041790642, 3.03466553655436401605943432713, 3.17462537696018144300319306397, 3.30410531069305064906965322222, 3.38627102339797634936611315838, 3.40396672457536603883092739752, 3.72282298382036659245105463868, 3.75033818031506282456090147919, 3.93483995836902272256927728580

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

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