# Properties

 Label 16-3200e8-1.1-c1e8-0-0 Degree $16$ Conductor $1.100\times 10^{28}$ Sign $1$ Analytic cond. $1.81725\times 10^{11}$ Root an. cond. $5.05491$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 4·9-s + 24·41-s + 56·49-s + 22·81-s − 72·89-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
 L(s)  = 1 − 4/3·9-s + 3.74·41-s + 8·49-s + 22/9·81-s − 7.63·89-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-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(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{56} \cdot 5^{16}$$ Sign: $1$ Analytic conductor: $$1.81725\times 10^{11}$$ Root analytic conductor: $$5.05491$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{3200} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{56} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6483129675$$ $$L(\frac12)$$ $$\approx$$ $$0.6483129675$$ $$L(\frac{3}{2})$$ 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$$
good3 $$( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
7 $$( 1 - p T^{2} )^{8}$$
11 $$( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
13 $$( 1 + p T^{2} )^{8}$$
17 $$( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
19 $$( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
23 $$( 1 - p T^{2} )^{8}$$
29 $$( 1 - p T^{2} )^{8}$$
31 $$( 1 + p T^{2} )^{8}$$
37 $$( 1 + p T^{2} )^{8}$$
41 $$( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4}$$
43 $$( 1 + 14 T^{2} + p^{2} T^{4} )^{4}$$
47 $$( 1 - p T^{2} )^{8}$$
53 $$( 1 + p T^{2} )^{8}$$
59 $$( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4}$$
61 $$( 1 - p T^{2} )^{8}$$
67 $$( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
71 $$( 1 + p T^{2} )^{8}$$
73 $$( 1 + 142 T^{2} + 14835 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
79 $$( 1 + p T^{2} )^{8}$$
83 $$( 1 - 158 T^{2} + 18075 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
89 $$( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{4}$$
97 $$( 1 - 94 T^{2} + p^{2} 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

−3.70056966729130069987874686345, −3.49316291295560357471349108757, −3.34091203388788965352702552212, −3.23254197082269130964774556293, −3.03850540824081876243881689517, −2.85142967645754438315352807658, −2.77953757995571206974801453813, −2.67591513594408314304671028176, −2.66504344399960320604746914705, −2.63692331324075320031265688784, −2.46096172149270170728816452702, −2.20462981083968075415354924776, −2.16203483443736676637964150574, −2.11608160927965895114788012817, −2.09856075260930554006508240854, −1.71312540201654906595398209230, −1.37070245846762250700265940440, −1.35739361210159566765396160787, −1.27424355067160224852716213942, −1.04915517856324537507270032164, −0.964447248041121248330458733957, −0.822448319716665190293184541911, −0.58282748740622291405485056933, −0.35822190437345616787880044762, −0.06695356096495571207658588099, 0.06695356096495571207658588099, 0.35822190437345616787880044762, 0.58282748740622291405485056933, 0.822448319716665190293184541911, 0.964447248041121248330458733957, 1.04915517856324537507270032164, 1.27424355067160224852716213942, 1.35739361210159566765396160787, 1.37070245846762250700265940440, 1.71312540201654906595398209230, 2.09856075260930554006508240854, 2.11608160927965895114788012817, 2.16203483443736676637964150574, 2.20462981083968075415354924776, 2.46096172149270170728816452702, 2.63692331324075320031265688784, 2.66504344399960320604746914705, 2.67591513594408314304671028176, 2.77953757995571206974801453813, 2.85142967645754438315352807658, 3.03850540824081876243881689517, 3.23254197082269130964774556293, 3.34091203388788965352702552212, 3.49316291295560357471349108757, 3.70056966729130069987874686345

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

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