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Properties

Label 16-31e16-1.1-c0e8-0-0
Degree $16$
Conductor $7.274\times 10^{23}$
Sign $1$
Analytic cond. $0.00279926$
Root an. cond. $0.692532$
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  + 2·2-s + 3·4-s + 4·5-s − 7-s + 2·8-s + 9-s + 8·10-s − 2·14-s + 16-s + 2·18-s − 19-s + 12·20-s + 10·25-s − 3·28-s − 2·32-s − 4·35-s + 3·36-s − 2·38-s + 8·40-s − 41-s + 4·45-s − 4·47-s + 20·50-s − 2·56-s − 59-s − 63-s − 4·64-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·5-s − 7-s + 2·8-s + 9-s + 8·10-s − 2·14-s + 16-s + 2·18-s − 19-s + 12·20-s + 10·25-s − 3·28-s − 2·32-s − 4·35-s + 3·36-s − 2·38-s + 8·40-s − 41-s + 4·45-s − 4·47-s + 20·50-s − 2·56-s − 59-s − 63-s − 4·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(31^{16}\)
Sign: $1$
Analytic conductor: \(0.00279926\)
Root analytic conductor: \(0.692532\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 31^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.233665649\)
\(L(\frac12)\) \(\approx\) \(5.233665649\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
3 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
5 \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
61 \( ( 1 - T )^{8}( 1 + T )^{8} \)
67 \( ( 1 + T + T^{2} )^{8} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
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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

−4.55495391260238007933615446520, −4.47662477328812333837678216858, −4.37816571753116951183572140188, −4.26716324177162205069320005051, −4.17877722064208209460311117332, −3.91734267389053394036883341751, −3.66391454847259638094659734680, −3.44368012399921943526748564184, −3.37894128414279609518809411443, −3.17670698977402126135125818953, −3.12073496653533552091809481045, −3.09585172393795666832028984302, −3.03803085588313957998326853375, −2.97479553097848224677383139969, −2.71665754552043616045680626578, −2.55025535752551345783689703886, −2.27960521838163076910440479858, −2.09545288495577375716758236149, −1.91889955753860796018432583885, −1.82322750323479001888303308449, −1.81280927561202971359910187959, −1.62630539029025569950701040810, −1.54376595017761818053877068744, −1.16416891651765383511647796994, −1.00061971667030243643718911363, 1.00061971667030243643718911363, 1.16416891651765383511647796994, 1.54376595017761818053877068744, 1.62630539029025569950701040810, 1.81280927561202971359910187959, 1.82322750323479001888303308449, 1.91889955753860796018432583885, 2.09545288495577375716758236149, 2.27960521838163076910440479858, 2.55025535752551345783689703886, 2.71665754552043616045680626578, 2.97479553097848224677383139969, 3.03803085588313957998326853375, 3.09585172393795666832028984302, 3.12073496653533552091809481045, 3.17670698977402126135125818953, 3.37894128414279609518809411443, 3.44368012399921943526748564184, 3.66391454847259638094659734680, 3.91734267389053394036883341751, 4.17877722064208209460311117332, 4.26716324177162205069320005051, 4.37816571753116951183572140188, 4.47662477328812333837678216858, 4.55495391260238007933615446520

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

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