Properties

Label 16-3200e8-1.1-c0e8-0-2
Degree $16$
Conductor $1.100\times 10^{28}$
Sign $1$
Analytic cond. $42.3113$
Root an. cond. $1.26372$
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  − 2·13-s − 2·17-s + 25-s + 4·29-s − 2·37-s + 2·53-s + 2·73-s + 81-s + 10·89-s + 2·97-s − 8·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·13-s − 2·17-s + 25-s + 4·29-s − 2·37-s + 2·53-s + 2·73-s + 81-s + 10·89-s + 2·97-s − 8·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(42.3113\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 5^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.736635031\)
\(L(\frac12)\) \(\approx\) \(1.736635031\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
good3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
23 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
53 \( ( 1 + T^{2} )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
89 \( ( 1 - T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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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

−3.91774638700157139182497981357, −3.58691417295652220641385373025, −3.45231934125202235594437846661, −3.44815721012636015932098855729, −3.38283107348994227472426037544, −3.27105938648442573324135962377, −3.06768138586088866772665989695, −3.03438608557176455827680543950, −2.73289589462564306078027304816, −2.67611476656966156894670040029, −2.56391745628400610134957330604, −2.49769338831887000587190540143, −2.46754587853950075136474218024, −2.18147198244407397291609807311, −2.12193834580992001713722763947, −2.10219786304855690986400064380, −1.92489671041596882471779467117, −1.76960363162539352839939916410, −1.60842696854187760692360529991, −1.32363015267642575494438961262, −1.13245883968908674776834178519, −1.00176787655974913117146060524, −0.856932412507434451914351824898, −0.61201028299603333833511605368, −0.44826297439707773287003253848, 0.44826297439707773287003253848, 0.61201028299603333833511605368, 0.856932412507434451914351824898, 1.00176787655974913117146060524, 1.13245883968908674776834178519, 1.32363015267642575494438961262, 1.60842696854187760692360529991, 1.76960363162539352839939916410, 1.92489671041596882471779467117, 2.10219786304855690986400064380, 2.12193834580992001713722763947, 2.18147198244407397291609807311, 2.46754587853950075136474218024, 2.49769338831887000587190540143, 2.56391745628400610134957330604, 2.67611476656966156894670040029, 2.73289589462564306078027304816, 3.03438608557176455827680543950, 3.06768138586088866772665989695, 3.27105938648442573324135962377, 3.38283107348994227472426037544, 3.44815721012636015932098855729, 3.45231934125202235594437846661, 3.58691417295652220641385373025, 3.91774638700157139182497981357

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

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