Properties

Label 16-2944e8-1.1-c0e8-0-0
Degree $16$
Conductor $5.643\times 10^{27}$
Sign $1$
Analytic cond. $21.7149$
Root an. cond. $1.21212$
Motivic weight $0$
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  − 4·3-s + 6·9-s + 4·71-s − 14·81-s + ⋯
L(s)  = 1  − 4·3-s + 6·9-s + 4·71-s − 14·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(21.7149\)
Root analytic conductor: \(1.21212\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 23^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08188463861\)
\(L(\frac12)\) \(\approx\) \(0.08188463861\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( ( 1 + T^{4} )^{2} \)
good3 \( ( 1 + T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \)
5 \( ( 1 + T^{8} )^{2} \)
7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 + T^{8} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
17 \( ( 1 + T^{2} )^{8} \)
19 \( ( 1 + T^{8} )^{2} \)
29 \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
31 \( ( 1 - T^{4} + T^{8} )^{2} \)
37 \( ( 1 + T^{8} )^{2} \)
41 \( ( 1 - T^{4} + T^{8} )^{2} \)
43 \( ( 1 + T^{8} )^{2} \)
47 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
53 \( ( 1 + T^{8} )^{2} \)
59 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
61 \( ( 1 + T^{8} )^{2} \)
67 \( ( 1 + T^{8} )^{2} \)
71 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
73 \( ( 1 - T^{4} + T^{8} )^{2} \)
79 \( ( 1 + T^{2} )^{8} \)
83 \( ( 1 + T^{8} )^{2} \)
89 \( ( 1 + T^{4} )^{4} \)
97 \( ( 1 - T )^{8}( 1 + T )^{8} \)
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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

−3.96385539912643238575429334288, −3.65566816140971806812012077832, −3.51647689832121372778137264505, −3.51267640774563649995383335386, −3.49759599580125407385389036862, −3.36800786433194109780858918273, −3.15979953612996775129929388046, −3.05309382293484655470873046440, −2.90753691545783416261083772644, −2.79885012124362751162025734784, −2.55369672669792154335261219887, −2.46370679044555279447331332916, −2.43496032247245995113950895182, −2.13969590553772873439681372027, −2.11610748683573962909039842982, −2.11350363055892620864473025417, −1.97115293863514425032760795185, −1.40200946458726651335114168126, −1.33900248581301522685122783136, −1.20280810824941493860181141491, −1.17932900011014017221187757045, −1.07115398449624239043312101987, −0.75911349858312888171618783459, −0.54308230710794059643874043967, −0.19834164372945376931499299640, 0.19834164372945376931499299640, 0.54308230710794059643874043967, 0.75911349858312888171618783459, 1.07115398449624239043312101987, 1.17932900011014017221187757045, 1.20280810824941493860181141491, 1.33900248581301522685122783136, 1.40200946458726651335114168126, 1.97115293863514425032760795185, 2.11350363055892620864473025417, 2.11610748683573962909039842982, 2.13969590553772873439681372027, 2.43496032247245995113950895182, 2.46370679044555279447331332916, 2.55369672669792154335261219887, 2.79885012124362751162025734784, 2.90753691545783416261083772644, 3.05309382293484655470873046440, 3.15979953612996775129929388046, 3.36800786433194109780858918273, 3.49759599580125407385389036862, 3.51267640774563649995383335386, 3.51647689832121372778137264505, 3.65566816140971806812012077832, 3.96385539912643238575429334288

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

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