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Properties

Label 16-33e16-1.1-c0e8-0-2
Degree $16$
Conductor $1.978\times 10^{24}$
Sign $1$
Analytic cond. $0.00761167$
Root an. cond. $0.737212$
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  − 2·4-s + 16-s + 2·25-s + 4·97-s − 4·100-s + 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 2·4-s + 16-s + 2·25-s + 4·97-s − 4·100-s + 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(0.00761167\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4526177003\)
\(L(\frac12)\) \(\approx\) \(0.4526177003\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
7 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
13 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
23 \( ( 1 + T^{2} )^{8} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
67 \( ( 1 + T^{2} )^{8} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
79 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
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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

−4.62295279559899667408415720097, −4.36092048291769080420979355836, −4.09836508095924963361008301762, −3.98516837862355863464745061497, −3.95937128375413067679711817758, −3.90734582682606064973369848354, −3.89275023384104356986805901376, −3.66785063532021729064990312177, −3.39370624573737144402028576322, −3.17705554271808749388478793727, −3.09317449994281244609463585816, −3.05361966177090638218079065431, −3.00389255426501251501802060626, −2.90227370705713524909050056627, −2.51332928671590422618556932016, −2.31917304404941302629406842304, −2.26437847382682777689274174188, −2.14535273795654806424150270438, −1.94848483513618685276119927009, −1.61142701254934689006492194806, −1.56006069287549804082634985708, −1.44427171151413130515009200183, −0.876610300552967539927896933030, −0.76198671179119282246770457432, −0.73652379186982093346515039048, 0.73652379186982093346515039048, 0.76198671179119282246770457432, 0.876610300552967539927896933030, 1.44427171151413130515009200183, 1.56006069287549804082634985708, 1.61142701254934689006492194806, 1.94848483513618685276119927009, 2.14535273795654806424150270438, 2.26437847382682777689274174188, 2.31917304404941302629406842304, 2.51332928671590422618556932016, 2.90227370705713524909050056627, 3.00389255426501251501802060626, 3.05361966177090638218079065431, 3.09317449994281244609463585816, 3.17705554271808749388478793727, 3.39370624573737144402028576322, 3.66785063532021729064990312177, 3.89275023384104356986805901376, 3.90734582682606064973369848354, 3.95937128375413067679711817758, 3.98516837862355863464745061497, 4.09836508095924963361008301762, 4.36092048291769080420979355836, 4.62295279559899667408415720097

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

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