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Properties

Label 16-543e8-1.1-c0e8-0-0
Degree $16$
Conductor $7.558\times 10^{21}$
Sign $1$
Analytic cond. $2.90841\times 10^{-5}$
Root an. cond. $0.520569$
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  + 3-s + 4-s − 4·7-s + 9-s + 12-s + 2·13-s + 16-s − 8·19-s − 4·21-s − 2·25-s − 4·28-s − 3·31-s + 36-s − 37-s + 2·39-s − 43-s + 48-s + 6·49-s + 2·52-s − 8·57-s − 61-s − 4·63-s − 3·67-s − 73-s − 2·75-s − 8·76-s − 79-s + ⋯
L(s)  = 1  + 3-s + 4-s − 4·7-s + 9-s + 12-s + 2·13-s + 16-s − 8·19-s − 4·21-s − 2·25-s − 4·28-s − 3·31-s + 36-s − 37-s + 2·39-s − 43-s + 48-s + 6·49-s + 2·52-s − 8·57-s − 61-s − 4·63-s − 3·67-s − 73-s − 2·75-s − 8·76-s − 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 181^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 181^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 181^{8}\)
Sign: $1$
Analytic conductor: \(2.90841\times 10^{-5}\)
Root analytic conductor: \(0.520569\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 181^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1295848364\)
\(L(\frac12)\) \(\approx\) \(0.1295848364\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
181 \( ( 1 - T )^{8} \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
17 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
19 \( ( 1 + T + T^{2} )^{8} \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
53 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
67 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
83 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 - T )^{8}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
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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

−5.14207246474751943453192481082, −4.65241528451759909174092446839, −4.59115099869309800483750310660, −4.41667822937269181675439396199, −4.22398736989866395420628146404, −4.19100304286947862889215287396, −4.11996956234850002712460682487, −4.07811140718806225729177983763, −3.91107939313900031394143745498, −3.57263119999502576249070524756, −3.51926438510142504294353330787, −3.34188673129648605198750803408, −3.30224990932761622914119736111, −3.23482849299449705110805911520, −3.08796819327776069508675268512, −2.82465466954817416048684353405, −2.59696715890712543547739510062, −2.40991540468910821422691744282, −2.15341064005876628691796219456, −2.03455013292872581417983354635, −1.83401972899337582008840640443, −1.78908837971290193704265061740, −1.74785584544847567875651049809, −1.51298619536651392203905625496, −0.38241786401894814712775355814, 0.38241786401894814712775355814, 1.51298619536651392203905625496, 1.74785584544847567875651049809, 1.78908837971290193704265061740, 1.83401972899337582008840640443, 2.03455013292872581417983354635, 2.15341064005876628691796219456, 2.40991540468910821422691744282, 2.59696715890712543547739510062, 2.82465466954817416048684353405, 3.08796819327776069508675268512, 3.23482849299449705110805911520, 3.30224990932761622914119736111, 3.34188673129648605198750803408, 3.51926438510142504294353330787, 3.57263119999502576249070524756, 3.91107939313900031394143745498, 4.07811140718806225729177983763, 4.11996956234850002712460682487, 4.19100304286947862889215287396, 4.22398736989866395420628146404, 4.41667822937269181675439396199, 4.59115099869309800483750310660, 4.65241528451759909174092446839, 5.14207246474751943453192481082

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

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