Properties

Label 16-465e8-1.1-c0e8-0-1
Degree $16$
Conductor $2.186\times 10^{21}$
Sign $1$
Analytic cond. $8.41163\times 10^{-6}$
Root an. cond. $0.481731$
Motivic weight $0$
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  − 2·2-s − 3-s + 3·4-s + 4·5-s + 2·6-s − 2·8-s + 9-s − 8·10-s − 3·12-s − 4·15-s + 16-s + 17-s − 2·18-s − 6·19-s + 12·20-s − 6·23-s + 2·24-s + 6·25-s + 8·30-s − 2·31-s + 2·32-s − 2·34-s + 3·36-s + 12·38-s − 8·40-s + 4·45-s + 12·46-s + ⋯
L(s)  = 1  − 2·2-s − 3-s + 3·4-s + 4·5-s + 2·6-s − 2·8-s + 9-s − 8·10-s − 3·12-s − 4·15-s + 16-s + 17-s − 2·18-s − 6·19-s + 12·20-s − 6·23-s + 2·24-s + 6·25-s + 8·30-s − 2·31-s + 2·32-s − 2·34-s + 3·36-s + 12·38-s − 8·40-s + 4·45-s + 12·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 31^{8}\)
Sign: $1$
Analytic conductor: \(8.41163\times 10^{-6}\)
Root analytic conductor: \(0.481731\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 5^{8} \cdot 31^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1682578135\)
\(L(\frac12)\) \(\approx\) \(0.1682578135\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \)
5 \( ( 1 - T + T^{2} )^{4} \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
good2 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
7 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
11 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
19 \( ( 1 + T + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
43 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
47 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
53 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
67 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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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

−5.25742250890246169110994947315, −5.11240298815598550152785122659, −4.82764859226951970867722484627, −4.82402797145778436198798470467, −4.51729630872309131675704942683, −4.46981470919685683800518437997, −4.44755270965318884367978486224, −4.07784348472268510782985739312, −3.96048637845052247290497248046, −3.95885014133805762819881333009, −3.75548932802332619620886991724, −3.55121191141590998400153094158, −3.35485337814215447136528948385, −3.14138438729138123947847482513, −2.80991543779246455979272083476, −2.40866207458106990006195841559, −2.31613845537286452056650186894, −2.22266823897879628221397964537, −2.19420369782610027285478852038, −2.01598465789310188260066149448, −1.84485280034613497234783018436, −1.74195498422415665342088143539, −1.68207407275644732539480900222, −1.54647328164570292722048344978, −0.979127647107487759404565963560, 0.979127647107487759404565963560, 1.54647328164570292722048344978, 1.68207407275644732539480900222, 1.74195498422415665342088143539, 1.84485280034613497234783018436, 2.01598465789310188260066149448, 2.19420369782610027285478852038, 2.22266823897879628221397964537, 2.31613845537286452056650186894, 2.40866207458106990006195841559, 2.80991543779246455979272083476, 3.14138438729138123947847482513, 3.35485337814215447136528948385, 3.55121191141590998400153094158, 3.75548932802332619620886991724, 3.95885014133805762819881333009, 3.96048637845052247290497248046, 4.07784348472268510782985739312, 4.44755270965318884367978486224, 4.46981470919685683800518437997, 4.51729630872309131675704942683, 4.82402797145778436198798470467, 4.82764859226951970867722484627, 5.11240298815598550152785122659, 5.25742250890246169110994947315

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

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