Properties

Label 16-429e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.147\times 10^{21}$
Sign $1$
Analytic cond. $4.41484\times 10^{-6}$
Root an. cond. $0.462708$
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  + 2·3-s − 3·4-s + 9-s − 6·12-s + 2·13-s + 6·16-s + 2·25-s − 3·36-s + 4·39-s − 4·43-s + 12·48-s − 2·49-s − 6·52-s − 6·61-s − 10·64-s + 4·75-s − 6·79-s − 6·100-s + 6·103-s + 2·117-s + 121-s + 127-s − 8·129-s + 131-s + 137-s + 139-s + 6·144-s + ⋯
L(s)  = 1  + 2·3-s − 3·4-s + 9-s − 6·12-s + 2·13-s + 6·16-s + 2·25-s − 3·36-s + 4·39-s − 4·43-s + 12·48-s − 2·49-s − 6·52-s − 6·61-s − 10·64-s + 4·75-s − 6·79-s − 6·100-s + 6·103-s + 2·117-s + 121-s + 127-s − 8·129-s + 131-s + 137-s + 139-s + 6·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 11^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(4.41484\times 10^{-6}\)
Root analytic conductor: \(0.462708\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{429} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 11^{8} \cdot 13^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2464889815\)
\(L(\frac12)\) \(\approx\) \(0.2464889815\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
11 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
good2 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 - T )^{8}( 1 + T )^{8} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
89 \( ( 1 + T^{2} )^{8} \)
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

−5.08341753640254701040434817044, −4.91491735992342149852661728556, −4.88767612510308161591205278996, −4.78666277226373006225493738430, −4.67677027902836269375126871237, −4.54004502837107880212446274385, −4.38133279910081686622752697949, −4.05018168884220264028134805243, −4.04916246036323849635359222255, −3.99226320678547777366318435248, −3.67596515694170709200913260421, −3.55073369425341405866554275408, −3.30515647267295733169903165810, −3.16965587422446130209045635720, −3.13907700815883792100299171842, −3.06899231554797084728625985944, −3.05892434731011445562152044693, −2.77380936128371061368570598996, −2.64677487073326551684795938580, −2.01248883791131144715996958368, −1.90755163675949141804022711470, −1.77684000250295276032352506307, −1.30437242615509866641677873868, −1.29084557054643480212245870162, −1.20107535848543875899676196178, 1.20107535848543875899676196178, 1.29084557054643480212245870162, 1.30437242615509866641677873868, 1.77684000250295276032352506307, 1.90755163675949141804022711470, 2.01248883791131144715996958368, 2.64677487073326551684795938580, 2.77380936128371061368570598996, 3.05892434731011445562152044693, 3.06899231554797084728625985944, 3.13907700815883792100299171842, 3.16965587422446130209045635720, 3.30515647267295733169903165810, 3.55073369425341405866554275408, 3.67596515694170709200913260421, 3.99226320678547777366318435248, 4.04916246036323849635359222255, 4.05018168884220264028134805243, 4.38133279910081686622752697949, 4.54004502837107880212446274385, 4.67677027902836269375126871237, 4.78666277226373006225493738430, 4.88767612510308161591205278996, 4.91491735992342149852661728556, 5.08341753640254701040434817044

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

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