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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

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.