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Properties

Label 16-2400e8-1.1-c0e8-0-2
Degree $16$
Conductor $1.101\times 10^{27}$
Sign $1$
Analytic cond. $4.23591$
Root an. cond. $1.09442$
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  + 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + 257-s + ⋯
L(s)  = 1  + 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + 257-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(4.23591\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.654471690\)
\(L(\frac12)\) \(\approx\) \(1.654471690\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T^{4} + T^{8} \)
5 \( 1 \)
good7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} )^{4} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T )^{8}( 1 + T )^{8} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{2} )^{8} \)
61 \( ( 1 - T )^{8}( 1 + T )^{8} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 - T^{4} + T^{8} )^{2} \)
79 \( ( 1 + T^{2} )^{8} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 - T^{2} + T^{4} )^{4} \)
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

−3.95262560438399134076474247375, −3.91742498726915109474744118968, −3.79909814117354294672876238443, −3.55139343666963079446374965515, −3.40077123692003507962726773793, −3.38627239398283981752270106147, −3.24020540286045557931749766086, −3.22202114026809409496521391928, −3.00047351197616449221700554021, −2.81261766276377231908123865275, −2.69703380465988015137939010187, −2.58855474464376291379027404222, −2.57505236827649852067124662268, −2.44715934206757391219450084438, −2.17493723178632031920049787667, −2.02698487078208416048280700495, −1.92069277720190121828216721513, −1.87459035048984399193740352005, −1.43504208371154466812001841293, −1.43297833105325966581701792241, −1.42053107648604393934414043739, −1.34080817360016857976838876563, −0.72984464960770401999361856084, −0.68730169378393681016313771010, −0.55239926231527711127773673288, 0.55239926231527711127773673288, 0.68730169378393681016313771010, 0.72984464960770401999361856084, 1.34080817360016857976838876563, 1.42053107648604393934414043739, 1.43297833105325966581701792241, 1.43504208371154466812001841293, 1.87459035048984399193740352005, 1.92069277720190121828216721513, 2.02698487078208416048280700495, 2.17493723178632031920049787667, 2.44715934206757391219450084438, 2.57505236827649852067124662268, 2.58855474464376291379027404222, 2.69703380465988015137939010187, 2.81261766276377231908123865275, 3.00047351197616449221700554021, 3.22202114026809409496521391928, 3.24020540286045557931749766086, 3.38627239398283981752270106147, 3.40077123692003507962726773793, 3.55139343666963079446374965515, 3.79909814117354294672876238443, 3.91742498726915109474744118968, 3.95262560438399134076474247375

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

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