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Properties

Label 16-651e8-1.1-c0e8-0-0
Degree $16$
Conductor $3.226\times 10^{22}$
Sign $1$
Analytic cond. $0.000124138$
Root an. cond. $0.569992$
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  − 2·3-s + 4-s − 7-s + 9-s − 2·12-s + 13-s + 16-s + 2·21-s − 8·25-s − 28-s − 31-s + 36-s + 3·37-s − 2·39-s − 5·43-s − 2·48-s + 49-s + 52-s + 2·61-s − 63-s + 67-s − 73-s + 16·75-s − 2·79-s + 2·84-s − 91-s + 2·93-s + ⋯
L(s)  = 1  − 2·3-s + 4-s − 7-s + 9-s − 2·12-s + 13-s + 16-s + 2·21-s − 8·25-s − 28-s − 31-s + 36-s + 3·37-s − 2·39-s − 5·43-s − 2·48-s + 49-s + 52-s + 2·61-s − 63-s + 67-s − 73-s + 16·75-s − 2·79-s + 2·84-s − 91-s + 2·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 7^{8} \cdot 31^{8}\)
Sign: $1$
Analytic conductor: \(0.000124138\)
Root analytic conductor: \(0.569992\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 7^{8} \cdot 31^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07830178488\)
\(L(\frac12)\) \(\approx\) \(0.07830178488\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7 \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \)
31 \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \)
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 \( ( 1 + T^{2} )^{8} \)
11 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
23 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
29 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
41 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
43 \( ( 1 + T + T^{2} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
47 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
53 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
79 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
97 \( ( 1 + T + T^{2} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
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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

−5.12575272077948439768520545609, −4.55436590653883558039446470200, −4.46722183974658321599090360415, −4.44855560745142449581421036003, −4.33796815582767955250492189968, −4.06429979470476895484924199903, −4.02534695338517359404518679576, −3.97336186919728612823989764482, −3.74885049385110226276987567578, −3.70168269131654433246116271288, −3.49673447100127983341624560276, −3.43964702152219192528045791733, −2.99451864133082306711841498771, −2.97357804455718438916504704335, −2.90816659943790793453485417432, −2.84712714102701571400045086997, −2.36188041589498580223174952639, −2.30491786140114385542705935895, −1.94372654216960986925790702935, −1.87042952854743141367267594926, −1.78775132148771652639996393074, −1.74363270228082841656787205857, −1.37858950131948021447132864807, −1.10266904532117267014344145942, −0.35256402371821093890312635013, 0.35256402371821093890312635013, 1.10266904532117267014344145942, 1.37858950131948021447132864807, 1.74363270228082841656787205857, 1.78775132148771652639996393074, 1.87042952854743141367267594926, 1.94372654216960986925790702935, 2.30491786140114385542705935895, 2.36188041589498580223174952639, 2.84712714102701571400045086997, 2.90816659943790793453485417432, 2.97357804455718438916504704335, 2.99451864133082306711841498771, 3.43964702152219192528045791733, 3.49673447100127983341624560276, 3.70168269131654433246116271288, 3.74885049385110226276987567578, 3.97336186919728612823989764482, 4.02534695338517359404518679576, 4.06429979470476895484924199903, 4.33796815582767955250492189968, 4.44855560745142449581421036003, 4.46722183974658321599090360415, 4.55436590653883558039446470200, 5.12575272077948439768520545609

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

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