Properties

Label 16-340e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.786\times 10^{20}$
Sign $1$
Analytic cond. $6.87208\times 10^{-7}$
Root an. cond. $0.411924$
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  − 8·41-s − 8·53-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 + 239-s + 241-s + 251-s − 256-s + ⋯
L(s)  = 1  − 8·41-s − 8·53-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 + 239-s + 241-s + 251-s − 256-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(6.87208\times 10^{-7}\)
Root analytic conductor: \(0.411924\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{8} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1559106341\)
\(L(\frac12)\) \(\approx\) \(0.1559106341\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{8} \)
5 \( ( 1 + T^{4} )^{2} \)
17 \( ( 1 + T^{4} )^{2} \)
good3 \( 1 + T^{16} \)
7 \( 1 + T^{16} \)
11 \( 1 + T^{16} \)
13 \( ( 1 + T^{8} )^{2} \)
19 \( ( 1 + T^{8} )^{2} \)
23 \( 1 + T^{16} \)
29 \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \)
31 \( 1 + T^{16} \)
37 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
41 \( ( 1 + T )^{8}( 1 + T^{8} ) \)
43 \( ( 1 + T^{8} )^{2} \)
47 \( ( 1 - T )^{8}( 1 + T )^{8} \)
53 \( ( 1 + T )^{8}( 1 + T^{4} )^{2} \)
59 \( ( 1 + T^{8} )^{2} \)
61 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( 1 + T^{16} \)
73 \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \)
79 \( 1 + T^{16} \)
83 \( ( 1 + T^{8} )^{2} \)
89 \( ( 1 + T^{8} )^{2} \)
97 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
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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

−5.31783900364234095405911430687, −5.28111685145739033582250388689, −5.09920397082244867629315244748, −5.07206191199456102538309649866, −4.88791472869590959168353984990, −4.58453801510145606637910457357, −4.53095393717841102772367631615, −4.49719197952140931147390686853, −4.49562896134849485628360440663, −4.02248825259788465511878398465, −3.98697425693094947976893203521, −3.57875533179538101607825070442, −3.45627541871799978805835447456, −3.40804222518368976547361237403, −3.33591874500545902851532078475, −3.01154029374935342820264667856, −2.98025515857140904230058364974, −2.96564673644581019576631843891, −2.69337738051985714842305283815, −1.98289049071698134190499295737, −1.93425242901049769485208713414, −1.91562310815840845655103316997, −1.69630287614801232554884448219, −1.59141605119153413770659423581, −1.19241834251483778955582985724, 1.19241834251483778955582985724, 1.59141605119153413770659423581, 1.69630287614801232554884448219, 1.91562310815840845655103316997, 1.93425242901049769485208713414, 1.98289049071698134190499295737, 2.69337738051985714842305283815, 2.96564673644581019576631843891, 2.98025515857140904230058364974, 3.01154029374935342820264667856, 3.33591874500545902851532078475, 3.40804222518368976547361237403, 3.45627541871799978805835447456, 3.57875533179538101607825070442, 3.98697425693094947976893203521, 4.02248825259788465511878398465, 4.49562896134849485628360440663, 4.49719197952140931147390686853, 4.53095393717841102772367631615, 4.58453801510145606637910457357, 4.88791472869590959168353984990, 5.07206191199456102538309649866, 5.09920397082244867629315244748, 5.28111685145739033582250388689, 5.31783900364234095405911430687

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

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