# 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 $0$

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## 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

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