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Properties

Label 16-3060e8-1.1-c1e8-0-1
Degree $16$
Conductor $7.687\times 10^{27}$
Sign $1$
Analytic cond. $1.27053\times 10^{11}$
Root an. cond. $4.94309$
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  − 3·5-s − 6·7-s − 10·17-s − 6·23-s + 5·25-s + 18·35-s + 26·37-s + 7·49-s − 32·59-s − 14·73-s + 30·85-s − 36·89-s − 12·97-s − 28·101-s − 30·107-s − 18·113-s + 18·115-s + 60·119-s + 56·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1.34·5-s − 2.26·7-s − 2.42·17-s − 1.25·23-s + 25-s + 3.04·35-s + 4.27·37-s + 49-s − 4.16·59-s − 1.63·73-s + 3.25·85-s − 3.81·89-s − 1.21·97-s − 2.78·101-s − 2.90·107-s − 1.69·113-s + 1.67·115-s + 5.50·119-s + 5.09·121-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(1.27053\times 10^{11}\)
Root analytic conductor: \(4.94309\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.02860215871\)
\(L(\frac12)\) \(\approx\) \(0.02860215871\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + 3 T + 4 T^{2} + 17 T^{3} + 54 T^{4} + 17 p T^{5} + 4 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
17 \( 1 + 10 T + 64 T^{2} + 358 T^{3} + 1630 T^{4} + 358 p T^{5} + 64 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
good7 \( ( 1 + 3 T + 10 T^{2} + 31 T^{3} + 106 T^{4} + 31 p T^{5} + 10 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( 1 - 56 T^{2} + 1619 T^{4} - 30044 T^{6} + 391264 T^{8} - 30044 p^{2} T^{10} + 1619 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 41 T^{2} + 966 T^{4} - 18159 T^{6} + 272754 T^{8} - 18159 p^{2} T^{10} + 966 p^{4} T^{12} - 41 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 35 T^{2} - 112 T^{3} + 544 T^{4} - 112 p T^{5} + 35 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 3 T + 52 T^{2} + 175 T^{3} + 1590 T^{4} + 175 p T^{5} + 52 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 59 T^{2} + 110 p T^{4} - 103581 T^{6} + 3432562 T^{8} - 103581 p^{2} T^{10} + 110 p^{5} T^{12} - 59 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 - 72 T^{2} + 3052 T^{4} - 136376 T^{6} + 5033254 T^{8} - 136376 p^{2} T^{10} + 3052 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \)
37 \( ( 1 - 13 T + 138 T^{2} - 871 T^{3} + 5946 T^{4} - 871 p T^{5} + 138 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 88 T^{2} + 5827 T^{4} - 8404 p T^{6} + 15260784 T^{8} - 8404 p^{3} T^{10} + 5827 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 105 T^{2} + 6682 T^{4} - 396223 T^{6} + 19233226 T^{8} - 396223 p^{2} T^{10} + 6682 p^{4} T^{12} - 105 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 319 T^{2} + 46078 T^{4} - 3981281 T^{6} + 227147778 T^{8} - 3981281 p^{2} T^{10} + 46078 p^{4} T^{12} - 319 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 191 T^{2} + 15898 T^{4} - 724057 T^{6} + 29674314 T^{8} - 724057 p^{2} T^{10} + 15898 p^{4} T^{12} - 191 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 4 T + p T^{2} )^{8} \)
61 \( 1 - 332 T^{2} + 54964 T^{4} - 5793492 T^{6} + 420853174 T^{8} - 5793492 p^{2} T^{10} + 54964 p^{4} T^{12} - 332 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 408 T^{2} + 76780 T^{4} - 8897960 T^{6} + 708120262 T^{8} - 8897960 p^{2} T^{10} + 76780 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 156 T^{2} + 23044 T^{4} - 2163492 T^{6} + 181753782 T^{8} - 2163492 p^{2} T^{10} + 23044 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 + 7 T + 134 T^{2} + 25 T^{3} + 6370 T^{4} + 25 p T^{5} + 134 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 - 68 T^{2} + 15892 T^{4} - 1193148 T^{6} + 128732758 T^{8} - 1193148 p^{2} T^{10} + 15892 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 188 T^{2} + 28036 T^{4} - 3350244 T^{6} + 290083222 T^{8} - 3350244 p^{2} T^{10} + 28036 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 18 T + 392 T^{2} + 4414 T^{3} + 53902 T^{4} + 4414 p T^{5} + 392 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 6 T + 96 T^{2} + 10 T^{3} + 5918 T^{4} + 10 p T^{5} + 96 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−3.71551887966495993736258697231, −3.49183635096686669439618713385, −3.25638883591874603201284935236, −3.21687541171150721031939353841, −3.14158557168125224461695980188, −2.99414866794359061658922745602, −2.78978805893752916601488873618, −2.78392886685380572098963652798, −2.76782098707329565436232145980, −2.60365889724519604901749913893, −2.48647320078099161075337571028, −2.48641657193803356050987696221, −2.25789808719070814525925252415, −2.01019967445448812716137587586, −1.69100904125674636757292210399, −1.68772090464703071563351969242, −1.66294528309342435493696918672, −1.49917923373254669981007707183, −1.19131837203916736193656278627, −1.14372775157164806939149779756, −0.948302667267017043052612506633, −0.62267259593082949264537482804, −0.33385041236903648959897291146, −0.25164835109967587217025856310, −0.04245868393437283859293486970, 0.04245868393437283859293486970, 0.25164835109967587217025856310, 0.33385041236903648959897291146, 0.62267259593082949264537482804, 0.948302667267017043052612506633, 1.14372775157164806939149779756, 1.19131837203916736193656278627, 1.49917923373254669981007707183, 1.66294528309342435493696918672, 1.68772090464703071563351969242, 1.69100904125674636757292210399, 2.01019967445448812716137587586, 2.25789808719070814525925252415, 2.48641657193803356050987696221, 2.48647320078099161075337571028, 2.60365889724519604901749913893, 2.76782098707329565436232145980, 2.78392886685380572098963652798, 2.78978805893752916601488873618, 2.99414866794359061658922745602, 3.14158557168125224461695980188, 3.21687541171150721031939353841, 3.25638883591874603201284935236, 3.49183635096686669439618713385, 3.71551887966495993736258697231

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

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