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Properties

Label 16-2005e8-1.1-c0e8-0-0
Degree $16$
Conductor $2.612\times 10^{26}$
Sign $1$
Analytic cond. $1.00501$
Root an. cond. $1.00031$
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 + 2·4-s − 2·5-s + 2·7-s − 2·8-s + 4·10-s − 4·14-s + 16-s − 4·20-s + 25-s + 4·28-s + 2·32-s − 4·35-s + 4·40-s + 2·43-s + 2·47-s + 2·49-s − 2·50-s − 4·56-s − 4·64-s + 8·70-s + 2·73-s − 2·80-s − 4·81-s + 8·83-s − 4·86-s − 4·94-s + ⋯
L(s)  = 1  − 2·2-s + 2·4-s − 2·5-s + 2·7-s − 2·8-s + 4·10-s − 4·14-s + 16-s − 4·20-s + 25-s + 4·28-s + 2·32-s − 4·35-s + 4·40-s + 2·43-s + 2·47-s + 2·49-s − 2·50-s − 4·56-s − 4·64-s + 8·70-s + 2·73-s − 2·80-s − 4·81-s + 8·83-s − 4·86-s − 4·94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 401^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 401^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 401^{8}\)
Sign: $1$
Analytic conductor: \(1.00501\)
Root analytic conductor: \(1.00031\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 401^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3177916273\)
\(L(\frac12)\) \(\approx\) \(0.3177916273\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
401 \( ( 1 - T )^{8} \)
good2 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
3 \( ( 1 + T^{4} )^{4} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
31 \( ( 1 - T )^{8}( 1 + T )^{8} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{2} )^{8} \)
61 \( ( 1 - T )^{8}( 1 + T )^{8} \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
79 \( ( 1 + T^{2} )^{8} \)
83 \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−4.07715377850195046417527964031, −3.93110282302529875702259876435, −3.88003993591884957925208687080, −3.73771235595019358428299899923, −3.62026032227337253544220732873, −3.30393388211978263788813316756, −3.29839925249418440758827739939, −3.17767250583146589038094486352, −3.17308709664276454036710089712, −3.08161237588720439488492947179, −2.78341634728293976287828378773, −2.40979735187258661593364543451, −2.33289865574592616128810409188, −2.30715847758054973690222186602, −2.25864216647391895235963555492, −2.21564714874350241352572035978, −2.15253366149485497566265419712, −1.90724786213332537769753829899, −1.49397951306707298342375207183, −1.38964612056731521837406879468, −1.13706067130765989634132740158, −1.10989846738800845030877744023, −1.01019034506687226157094589892, −0.54471736165815280362200658107, −0.53657553587055374050787212647, 0.53657553587055374050787212647, 0.54471736165815280362200658107, 1.01019034506687226157094589892, 1.10989846738800845030877744023, 1.13706067130765989634132740158, 1.38964612056731521837406879468, 1.49397951306707298342375207183, 1.90724786213332537769753829899, 2.15253366149485497566265419712, 2.21564714874350241352572035978, 2.25864216647391895235963555492, 2.30715847758054973690222186602, 2.33289865574592616128810409188, 2.40979735187258661593364543451, 2.78341634728293976287828378773, 3.08161237588720439488492947179, 3.17308709664276454036710089712, 3.17767250583146589038094486352, 3.29839925249418440758827739939, 3.30393388211978263788813316756, 3.62026032227337253544220732873, 3.73771235595019358428299899923, 3.88003993591884957925208687080, 3.93110282302529875702259876435, 4.07715377850195046417527964031

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

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