Properties

Label 16-1200e8-1.1-c0e8-0-0
Degree $16$
Conductor $4.300\times 10^{24}$
Sign $1$
Analytic cond. $0.0165465$
Root an. cond. $0.773872$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·61-s − 2·81-s − 8·121-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 + ⋯
L(s)  = 1  − 8·61-s − 2·81-s − 8·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(0.0165465\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4277790724\)
\(L(\frac12)\) \(\approx\) \(0.4277790724\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{4} )^{2} \)
5 \( 1 \)
good7 \( ( 1 - T^{4} + T^{8} )^{2} \)
11 \( ( 1 + T^{2} )^{8} \)
13 \( ( 1 - T^{4} + T^{8} )^{2} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( ( 1 - T^{2} + T^{4} )^{4} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 + T^{2} )^{8} \)
31 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 - T )^{8}( 1 + T )^{8} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 - T )^{8}( 1 + T )^{8} \)
61 \( ( 1 + T + T^{2} )^{8} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 + T^{2} )^{8} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( ( 1 - T^{4} + T^{8} )^{2} \)
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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

−4.48889629249691705884979550839, −4.33958209335588569819012305705, −4.09364051827906017197568685505, −3.89276714175002455734770705118, −3.89028868211495220529802022498, −3.75335578319457017535164966779, −3.73483921743346343209832230530, −3.55209047825886709964711488501, −3.31768618661380884244617420655, −3.26709994203518284727793812159, −2.99361070775509986676520892073, −2.91158429638798735460069858219, −2.74830163598140355564122869026, −2.63825893309554279608357118557, −2.56986317391321156244771585004, −2.55360387859411942940465678206, −2.36489843690771134487902723627, −1.84433059421072079853506284368, −1.76470960263916003172936311193, −1.63706512570907164309288813831, −1.59523592017145420401827546357, −1.38555442302089114742771613235, −1.22464184173182622059227492847, −1.02774388645613217780550038082, −0.35899072273799486367585526515, 0.35899072273799486367585526515, 1.02774388645613217780550038082, 1.22464184173182622059227492847, 1.38555442302089114742771613235, 1.59523592017145420401827546357, 1.63706512570907164309288813831, 1.76470960263916003172936311193, 1.84433059421072079853506284368, 2.36489843690771134487902723627, 2.55360387859411942940465678206, 2.56986317391321156244771585004, 2.63825893309554279608357118557, 2.74830163598140355564122869026, 2.91158429638798735460069858219, 2.99361070775509986676520892073, 3.26709994203518284727793812159, 3.31768618661380884244617420655, 3.55209047825886709964711488501, 3.73483921743346343209832230530, 3.75335578319457017535164966779, 3.89028868211495220529802022498, 3.89276714175002455734770705118, 4.09364051827906017197568685505, 4.33958209335588569819012305705, 4.48889629249691705884979550839

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

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