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Properties

Label 16-3200e8-1.1-c0e8-0-1
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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 5^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7527650645\)
\(L(\frac12)\) \(\approx\) \(0.7527650645\)
\(L(1)\) 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

−3.81246579954198126158675274481, −3.80498330943995031609068761867, −3.77967290164660695972365977510, −3.37625404651925308108995833512, −3.29300853141968039400977298610, −3.24754200186197610132636050923, −3.19064604361024945934384682474, −3.07811874828624651049161962562, −3.02729866320802390578541883153, −2.63336358978007568798857050090, −2.47664765871470159988706684440, −2.35582403769640637779959003439, −2.31117296637407786983136546815, −2.26630454388658692102564065705, −2.23382739782826012951355945531, −2.01208108204101234619926427325, −1.93773845971010464790778987993, −1.63400017221592345329292108300, −1.51687531081595597247784871128, −1.37135749251995518460145877372, −1.19697567233760887196401608385, −1.11636081852424014317510412329, −0.896026578594829711902654803981, −0.72785290252308125464931330632, −0.22394509308416144064359145954, 0.22394509308416144064359145954, 0.72785290252308125464931330632, 0.896026578594829711902654803981, 1.11636081852424014317510412329, 1.19697567233760887196401608385, 1.37135749251995518460145877372, 1.51687531081595597247784871128, 1.63400017221592345329292108300, 1.93773845971010464790778987993, 2.01208108204101234619926427325, 2.23382739782826012951355945531, 2.26630454388658692102564065705, 2.31117296637407786983136546815, 2.35582403769640637779959003439, 2.47664765871470159988706684440, 2.63336358978007568798857050090, 3.02729866320802390578541883153, 3.07811874828624651049161962562, 3.19064604361024945934384682474, 3.24754200186197610132636050923, 3.29300853141968039400977298610, 3.37625404651925308108995833512, 3.77967290164660695972365977510, 3.80498330943995031609068761867, 3.81246579954198126158675274481

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

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