Properties

Label 16-33e16-1.1-c0e8-0-1
Degree $16$
Conductor $1.978\times 10^{24}$
Sign $1$
Analytic cond. $0.00761167$
Root an. cond. $0.737212$
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  + 3-s + 4-s − 5-s + 9-s + 12-s − 15-s + 16-s − 20-s − 8·23-s − 31-s + 36-s + 2·37-s − 45-s − 47-s + 48-s + 49-s + 2·53-s − 59-s − 60-s + 4·67-s − 8·69-s + 2·71-s − 80-s + 16·89-s − 8·92-s − 93-s − 97-s + ⋯
L(s)  = 1  + 3-s + 4-s − 5-s + 9-s + 12-s − 15-s + 16-s − 20-s − 8·23-s − 31-s + 36-s + 2·37-s − 45-s − 47-s + 48-s + 49-s + 2·53-s − 59-s − 60-s + 4·67-s − 8·69-s + 2·71-s − 80-s + 16·89-s − 8·92-s − 93-s − 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(0.00761167\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1089} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.046070384\)
\(L(\frac12)\) \(\approx\) \(1.046070384\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.50374501051364827306764548972, −4.03037914288316472416734418445, −4.02966983663376212602488140550, −4.02619895337222724978167091349, −3.92197280252426064046130537760, −3.91891488221525270477614763313, −3.68861938348009959087195304882, −3.68631195970493028714483764478, −3.41079661368730100747894974500, −3.32650842818118920857484555597, −3.27487907789363666989478891112, −3.12126964966874120600157595047, −2.93731962987430778313204365455, −2.39855207873369818293307235635, −2.34277045188507119603525639726, −2.24054406024181301185342300060, −2.23857340382195697926311143151, −2.20736062269649460467375918487, −2.07066668989673035872327052712, −1.96165119242506724347629611573, −1.76944399601493162697272776785, −1.53858145345521626864743837131, −1.06772997651965484254035321898, −0.897428320786826787973931314659, −0.65512044014318118303178133755, 0.65512044014318118303178133755, 0.897428320786826787973931314659, 1.06772997651965484254035321898, 1.53858145345521626864743837131, 1.76944399601493162697272776785, 1.96165119242506724347629611573, 2.07066668989673035872327052712, 2.20736062269649460467375918487, 2.23857340382195697926311143151, 2.24054406024181301185342300060, 2.34277045188507119603525639726, 2.39855207873369818293307235635, 2.93731962987430778313204365455, 3.12126964966874120600157595047, 3.27487907789363666989478891112, 3.32650842818118920857484555597, 3.41079661368730100747894974500, 3.68631195970493028714483764478, 3.68861938348009959087195304882, 3.91891488221525270477614763313, 3.92197280252426064046130537760, 4.02619895337222724978167091349, 4.02966983663376212602488140550, 4.03037914288316472416734418445, 4.50374501051364827306764548972

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

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