Properties

Label 16-320e8-1.1-c1e8-0-1
Degree $16$
Conductor $1.100\times 10^{20}$
Sign $1$
Analytic cond. $1817.25$
Root an. cond. $1.59850$
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  − 4·7-s + 24·17-s + 12·23-s − 24·41-s − 12·47-s + 8·49-s − 32·73-s + 96·79-s − 8·81-s − 32·97-s + 28·103-s − 96·119-s − 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 1.51·7-s + 5.82·17-s + 2.50·23-s − 3.74·41-s − 1.75·47-s + 8/7·49-s − 3.74·73-s + 10.8·79-s − 8/9·81-s − 3.24·97-s + 2.75·103-s − 8.80·119-s − 4.36·121-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 − 3.78·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1817.25\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 5^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.806275293\)
\(L(\frac12)\) \(\approx\) \(2.806275293\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - 34 T^{4} + p^{4} T^{8} \)
good3 \( 1 + 8 T^{4} + 94 T^{8} + 8 p^{4} T^{12} + p^{8} T^{16} \)
7 \( ( 1 + 2 T + 2 T^{2} - 6 T^{3} - 82 T^{4} - 6 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 24 T^{2} + 302 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 2 T + p T^{2} )^{4} \)
19 \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \)
23 \( ( 1 - 6 T + 18 T^{2} - 102 T^{3} + 542 T^{4} - 102 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 64 T^{2} + 2190 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 2338 T^{4} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
43 \( 1 + 2632 T^{4} + 3107358 T^{8} + 2632 p^{4} T^{12} + p^{8} T^{16} \)
47 \( ( 1 + 6 T + 18 T^{2} + 246 T^{3} + 3326 T^{4} + 246 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 2846 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 112 T^{2} + 9822 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( 1 - 56 T^{4} - 39549474 T^{8} - 56 p^{4} T^{12} + p^{8} T^{16} \)
71 \( ( 1 - 80 T^{2} + 4878 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 16 T + 128 T^{2} + 1008 T^{3} + 7838 T^{4} + 1008 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 12 T + p T^{2} )^{8} \)
83 \( 1 + 8200 T^{4} + 98353758 T^{8} + 8200 p^{4} T^{12} + p^{8} T^{16} \)
89 \( ( 1 - 116 T^{2} + 7110 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 16 T + 128 T^{2} + 1392 T^{3} + 15038 T^{4} + 1392 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−5.13438410011949035827611955586, −5.00300328504119628250658705528, −4.99797413585837926245795086769, −4.97633324809786849724108151787, −4.84907233032674634405910677312, −4.61285552829091278501594235652, −4.23458053509013375024023428511, −3.89852017623922189853597685217, −3.85544204918237329699425067864, −3.75433268999048296284672652914, −3.50072910890815620855250866064, −3.43520546188331798267294970685, −3.35546001902201512383705057784, −3.27082878960383257427154112392, −3.02196512345690803106487342514, −2.87327222323351124839965407479, −2.83380114410836083869967203692, −2.42332715057219706725686221249, −2.20688305333237280610987585714, −1.92554315144311601114371901541, −1.56856732259898553235872835770, −1.36439592433597741770647535212, −1.07028622373396869751093103667, −1.04673497159672659432434930475, −0.45278802364928996872331092140, 0.45278802364928996872331092140, 1.04673497159672659432434930475, 1.07028622373396869751093103667, 1.36439592433597741770647535212, 1.56856732259898553235872835770, 1.92554315144311601114371901541, 2.20688305333237280610987585714, 2.42332715057219706725686221249, 2.83380114410836083869967203692, 2.87327222323351124839965407479, 3.02196512345690803106487342514, 3.27082878960383257427154112392, 3.35546001902201512383705057784, 3.43520546188331798267294970685, 3.50072910890815620855250866064, 3.75433268999048296284672652914, 3.85544204918237329699425067864, 3.89852017623922189853597685217, 4.23458053509013375024023428511, 4.61285552829091278501594235652, 4.84907233032674634405910677312, 4.97633324809786849724108151787, 4.99797413585837926245795086769, 5.00300328504119628250658705528, 5.13438410011949035827611955586

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

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