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Properties

Label 16-720e8-1.1-c1e8-0-9
Degree $16$
Conductor $7.222\times 10^{22}$
Sign $1$
Analytic cond. $1.19364\times 10^{6}$
Root an. cond. $2.39775$
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  − 2·4-s + 4·8-s − 8·11-s + 8·13-s + 2·16-s − 8·17-s + 8·19-s − 24·29-s + 8·31-s − 8·32-s − 8·37-s + 16·44-s + 48·49-s − 16·52-s − 8·59-s − 16·61-s + 8·64-s + 16·68-s − 16·76-s − 40·79-s − 32·83-s − 32·88-s − 48·97-s + 32·101-s + 32·104-s + 16·109-s + 24·113-s + ⋯
L(s)  = 1  − 4-s + 1.41·8-s − 2.41·11-s + 2.21·13-s + 1/2·16-s − 1.94·17-s + 1.83·19-s − 4.45·29-s + 1.43·31-s − 1.41·32-s − 1.31·37-s + 2.41·44-s + 48/7·49-s − 2.21·52-s − 1.04·59-s − 2.04·61-s + 64-s + 1.94·68-s − 1.83·76-s − 4.50·79-s − 3.51·83-s − 3.41·88-s − 4.87·97-s + 3.18·101-s + 3.13·104-s + 1.53·109-s + 2.25·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.19364\times 10^{6}\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{16} \cdot 5^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.677182296\)
\(L(\frac12)\) \(\approx\) \(1.677182296\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T^{2} - p^{2} T^{3} + p T^{4} - p^{3} T^{5} + p^{3} T^{6} + p^{4} T^{8} \)
3 \( 1 \)
5 \( ( 1 + T^{4} )^{2} \)
good7 \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{4} \)
11 \( 1 + 8 T + 32 T^{2} + 72 T^{3} + 28 p T^{4} + 184 p T^{5} + 8928 T^{6} + 23016 T^{7} + 59398 T^{8} + 23016 p T^{9} + 8928 p^{2} T^{10} + 184 p^{4} T^{11} + 28 p^{5} T^{12} + 72 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 - 8 T + 32 T^{2} - 120 T^{3} + 668 T^{4} - 3400 T^{5} + 13024 T^{6} - 48504 T^{7} + 179494 T^{8} - 48504 p T^{9} + 13024 p^{2} T^{10} - 3400 p^{3} T^{11} + 668 p^{4} T^{12} - 120 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
17 \( ( 1 + 4 T + 56 T^{2} + 12 p T^{3} + 1334 T^{4} + 12 p^{2} T^{5} + 56 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 8 T + 32 T^{2} - 56 T^{3} + 452 T^{4} - 4792 T^{5} + 25440 T^{6} - 65608 T^{7} + 138918 T^{8} - 65608 p T^{9} + 25440 p^{2} T^{10} - 4792 p^{3} T^{11} + 452 p^{4} T^{12} - 56 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 56 T^{2} + 2396 T^{4} - 74760 T^{6} + 1907718 T^{8} - 74760 p^{2} T^{10} + 2396 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
29 \( 1 + 24 T + 288 T^{2} + 2584 T^{3} + 19780 T^{4} + 130408 T^{5} + 771680 T^{6} + 4333352 T^{7} + 23623974 T^{8} + 4333352 p T^{9} + 771680 p^{2} T^{10} + 130408 p^{3} T^{11} + 19780 p^{4} T^{12} + 2584 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
31 \( ( 1 - 4 T + 84 T^{2} - 284 T^{3} + 3258 T^{4} - 284 p T^{5} + 84 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 + 8 T + 32 T^{2} + 152 T^{3} - 612 T^{4} - 1624 T^{5} + 18144 T^{6} + 350712 T^{7} + 4521958 T^{8} + 350712 p T^{9} + 18144 p^{2} T^{10} - 1624 p^{3} T^{11} - 612 p^{4} T^{12} + 152 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 24 T^{2} + 3356 T^{4} - 13352 T^{6} + 5175942 T^{8} - 13352 p^{2} T^{10} + 3356 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 + 256 T^{3} + 420 T^{4} - 14592 T^{5} + 32768 T^{6} - 125440 T^{7} - 7198298 T^{8} - 125440 p T^{9} + 32768 p^{2} T^{10} - 14592 p^{3} T^{11} + 420 p^{4} T^{12} + 256 p^{5} T^{13} + p^{8} T^{16} \)
47 \( ( 1 + 36 T^{2} + 416 T^{3} - 1258 T^{4} + 416 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( 1 + 5540 T^{4} + 17031334 T^{8} + 5540 p^{4} T^{12} + p^{8} T^{16} \)
59 \( 1 + 8 T + 32 T^{2} + 1256 T^{3} + 11572 T^{4} + 57672 T^{5} + 879840 T^{6} + 7404968 T^{7} + 37955718 T^{8} + 7404968 p T^{9} + 879840 p^{2} T^{10} + 57672 p^{3} T^{11} + 11572 p^{4} T^{12} + 1256 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 16 T + 128 T^{2} + 560 T^{3} - 2052 T^{4} - 14512 T^{5} + 187264 T^{6} + 4035568 T^{7} + 46807718 T^{8} + 4035568 p T^{9} + 187264 p^{2} T^{10} - 14512 p^{3} T^{11} - 2052 p^{4} T^{12} + 560 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 740 T^{4} + 22343014 T^{8} + 740 p^{4} T^{12} + p^{8} T^{16} \)
71 \( 1 - 296 T^{2} + 48444 T^{4} - 5355544 T^{6} + 6173194 p T^{8} - 5355544 p^{2} T^{10} + 48444 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 408 T^{2} + 76348 T^{4} - 8922664 T^{6} + 749683526 T^{8} - 8922664 p^{2} T^{10} + 76348 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 20 T + 436 T^{2} + 5004 T^{3} + 56570 T^{4} + 5004 p T^{5} + 436 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 32 T + 512 T^{2} + 5984 T^{3} + 63460 T^{4} + 668256 T^{5} + 6796800 T^{6} + 63883808 T^{7} + 580255974 T^{8} + 63883808 p T^{9} + 6796800 p^{2} T^{10} + 668256 p^{3} T^{11} + 63460 p^{4} T^{12} + 5984 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 120 T^{2} + 8412 T^{4} + 825144 T^{6} - 118403962 T^{8} + 825144 p^{2} T^{10} + 8412 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 + 24 T + 508 T^{2} + 6952 T^{3} + 78998 T^{4} + 6952 p T^{5} + 508 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.74036022863379595836322883381, −4.20859546502324061033422252014, −4.06265486147379290265937342959, −3.94632705848938534669603291654, −3.90613339230569350149134308389, −3.90139838122999720295097886295, −3.86545464539103911847125862756, −3.85950415318122191492215960003, −3.63061979040357615351536242724, −3.26538918945077343971721388061, −2.85125417842960728101774545748, −2.83874680819482699383950297060, −2.77776121105689613553268834945, −2.63643136114201967280396452464, −2.63055621669646416508034814535, −2.53026191678520538622918158296, −2.12693671639983165285798582499, −1.69459199237485730929406520775, −1.63111810589162199181713262745, −1.57730217335111270129380724282, −1.46089977040879358691669163127, −1.31693599469307774864356098127, −0.797592002734954829268969664258, −0.42294246176367623962748546982, −0.29007301863299996637026392327, 0.29007301863299996637026392327, 0.42294246176367623962748546982, 0.797592002734954829268969664258, 1.31693599469307774864356098127, 1.46089977040879358691669163127, 1.57730217335111270129380724282, 1.63111810589162199181713262745, 1.69459199237485730929406520775, 2.12693671639983165285798582499, 2.53026191678520538622918158296, 2.63055621669646416508034814535, 2.63643136114201967280396452464, 2.77776121105689613553268834945, 2.83874680819482699383950297060, 2.85125417842960728101774545748, 3.26538918945077343971721388061, 3.63061979040357615351536242724, 3.85950415318122191492215960003, 3.86545464539103911847125862756, 3.90139838122999720295097886295, 3.90613339230569350149134308389, 3.94632705848938534669603291654, 4.06265486147379290265937342959, 4.20859546502324061033422252014, 4.74036022863379595836322883381

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

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