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Properties

Label 16-15e16-1.1-c3e8-0-0
Degree $16$
Conductor $6.568\times 10^{18}$
Sign $1$
Analytic cond. $9.64692\times 10^{8}$
Root an. cond. $3.64354$
Motivic weight $3$
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  − 15·4-s + 90·9-s + 74·11-s + 176·16-s − 140·19-s + 650·29-s + 24·31-s − 1.35e3·36-s − 476·41-s − 1.11e3·44-s − 1.31e3·49-s − 170·59-s + 494·61-s − 2.62e3·64-s + 1.57e3·71-s + 2.10e3·76-s − 1.68e3·79-s + 4.61e3·81-s + 4.26e3·89-s + 6.66e3·99-s − 3.63e3·101-s + 4.94e3·109-s − 9.75e3·116-s + 6.96e3·121-s − 360·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.87·4-s + 10/3·9-s + 2.02·11-s + 11/4·16-s − 1.69·19-s + 4.16·29-s + 0.139·31-s − 6.25·36-s − 1.81·41-s − 3.80·44-s − 3.83·49-s − 0.375·59-s + 1.03·61-s − 5.12·64-s + 2.63·71-s + 3.16·76-s − 2.39·79-s + 19/3·81-s + 5.07·89-s + 6.76·99-s − 3.58·101-s + 4.34·109-s − 7.80·116-s + 5.23·121-s − 0.260·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(9.64692\times 10^{8}\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 5^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.008875644\)
\(L(\frac12)\) \(\approx\) \(1.008875644\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - 5 p^{2} T^{2} + p^{6} T^{4} )^{2} \)
5 \( 1 \)
good2 \( 1 + 15 T^{2} + 49 T^{4} + 45 p^{4} T^{6} + 195 p^{6} T^{8} + 45 p^{10} T^{10} + 49 p^{12} T^{12} + 15 p^{18} T^{14} + p^{24} T^{16} \)
7 \( 1 + 1315 T^{2} + 1062289 T^{4} + 567603970 T^{6} + 227520748870 T^{8} + 567603970 p^{6} T^{10} + 1062289 p^{12} T^{12} + 1315 p^{18} T^{14} + p^{24} T^{16} \)
11 \( ( 1 - 37 T - 1429 T^{2} - 5032 T^{3} + 4235104 T^{4} - 5032 p^{3} T^{5} - 1429 p^{6} T^{6} - 37 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
13 \( 1 + 1460 T^{2} - 1431686 T^{4} - 8891884720 T^{6} - 20889904069085 T^{8} - 8891884720 p^{6} T^{10} - 1431686 p^{12} T^{12} + 1460 p^{18} T^{14} + p^{24} T^{16} \)
17 \( ( 1 - 375 p T^{2} + 27863888 T^{4} - 375 p^{7} T^{6} + p^{12} T^{8} )^{2} \)
19 \( ( 1 + 35 T + 8868 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
23 \( 1 + 4295 T^{2} + 28886509 T^{4} - 1316465870290 T^{6} - 24536463351114290 T^{8} - 1316465870290 p^{6} T^{10} + 28886509 p^{12} T^{12} + 4295 p^{18} T^{14} + p^{24} T^{16} \)
29 \( ( 1 - 325 T + 42989 T^{2} - 4503850 T^{3} + 752357050 T^{4} - 4503850 p^{3} T^{5} + 42989 p^{6} T^{6} - 325 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
31 \( ( 1 - 6 T - 29755 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
37 \( ( 1 + 2560 T^{2} + 3504222 p^{2} T^{4} + 2560 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
41 \( ( 1 + 238 T - 65659 T^{2} - 90202 p T^{3} + 3750124 p^{2} T^{4} - 90202 p^{4} T^{5} - 65659 p^{6} T^{6} + 238 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
43 \( 1 + 188435 T^{2} + 14575623769 T^{4} + 1562012968024730 T^{6} + \)\(17\!\cdots\!10\)\( T^{8} + 1562012968024730 p^{6} T^{10} + 14575623769 p^{12} T^{12} + 188435 p^{18} T^{14} + p^{24} T^{16} \)
47 \( 1 + 7395 T^{2} - 20707035791 T^{4} - 5891661886590 T^{6} + \)\(31\!\cdots\!90\)\( T^{8} - 5891661886590 p^{6} T^{10} - 20707035791 p^{12} T^{12} + 7395 p^{18} T^{14} + p^{24} T^{16} \)
53 \( ( 1 - 405420 T^{2} + 81280546358 T^{4} - 405420 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
59 \( ( 1 + 85 T - 385531 T^{2} - 1530170 T^{3} + 110592878620 T^{4} - 1530170 p^{3} T^{5} - 385531 p^{6} T^{6} + 85 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
61 \( ( 1 - 247 T - 94499 T^{2} + 73718138 T^{3} - 41185520126 T^{4} + 73718138 p^{3} T^{5} - 94499 p^{6} T^{6} - 247 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
67 \( 1 + 460810 T^{2} + 80916049369 T^{4} - 22804084934881670 T^{6} - \)\(10\!\cdots\!00\)\( T^{8} - 22804084934881670 p^{6} T^{10} + 80916049369 p^{12} T^{12} + 460810 p^{18} T^{14} + p^{24} T^{16} \)
71 \( ( 1 - 394 T + 654806 T^{2} - 394 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
73 \( ( 1 - 533795 T^{2} + 145866746628 T^{4} - 533795 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
79 \( ( 1 + 840 T - 409226 T^{2} + 108148320 T^{3} + 651861236355 T^{4} + 108148320 p^{3} T^{5} - 409226 p^{6} T^{6} + 840 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
83 \( 1 - 1173685 T^{2} + 632826673129 T^{4} - 106604704532594230 T^{6} - \)\(26\!\cdots\!70\)\( T^{8} - 106604704532594230 p^{6} T^{10} + 632826673129 p^{12} T^{12} - 1173685 p^{18} T^{14} + p^{24} T^{16} \)
89 \( ( 1 - 1065 T + 874888 T^{2} - 1065 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
97 \( 1 + 2836055 T^{2} + 4516954120609 T^{4} + 5275941002175278690 T^{6} + \)\(50\!\cdots\!90\)\( T^{8} + 5275941002175278690 p^{6} T^{10} + 4516954120609 p^{12} T^{12} + 2836055 p^{18} T^{14} + p^{24} T^{16} \)
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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

−4.97479886292797345066595949024, −4.81940249351946668265277639299, −4.63182455422714472819024824361, −4.54107532225363014044598852534, −4.44409740452467061348113582600, −4.22021399891771980943251916758, −4.18844254220483479293381316940, −4.04036953688974081777502482038, −3.75804069887898372324694383078, −3.58759277903379197853129192205, −3.39629351976193073771249994748, −3.26105161529071445484340379525, −3.20268148057777608424245311848, −3.06065652772854343428097976860, −2.43721871799943055814180465258, −2.38729560536778823213696214316, −2.09949758285996934477747988260, −1.81692273881956616180767485236, −1.59446258866937769319337507776, −1.40315306137830026438656330660, −1.36409920659621904443395869350, −0.866919529283465979061566506181, −0.850467030198802479377128230569, −0.67918816010795960701625069500, −0.083468554658726522300315065908, 0.083468554658726522300315065908, 0.67918816010795960701625069500, 0.850467030198802479377128230569, 0.866919529283465979061566506181, 1.36409920659621904443395869350, 1.40315306137830026438656330660, 1.59446258866937769319337507776, 1.81692273881956616180767485236, 2.09949758285996934477747988260, 2.38729560536778823213696214316, 2.43721871799943055814180465258, 3.06065652772854343428097976860, 3.20268148057777608424245311848, 3.26105161529071445484340379525, 3.39629351976193073771249994748, 3.58759277903379197853129192205, 3.75804069887898372324694383078, 4.04036953688974081777502482038, 4.18844254220483479293381316940, 4.22021399891771980943251916758, 4.44409740452467061348113582600, 4.54107532225363014044598852534, 4.63182455422714472819024824361, 4.81940249351946668265277639299, 4.97479886292797345066595949024

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

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