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Properties

Label 16-544e8-1.1-c0e8-0-0
Degree $16$
Conductor $7.670\times 10^{21}$
Sign $1$
Analytic cond. $2.95153\times 10^{-5}$
Root an. cond. $0.521048$
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  − 8·53-s − 8·73-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 + 257-s + ⋯
L(s)  = 1  − 8·53-s − 8·73-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 + 257-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(2.95153\times 10^{-5}\)
Root analytic conductor: \(0.521048\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3019897594\)
\(L(\frac12)\) \(\approx\) \(0.3019897594\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( ( 1 + T^{4} )^{2} \)
good3 \( 1 + T^{16} \)
5 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
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 \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \)
43 \( ( 1 + T^{8} )^{2} \)
47 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T )^{8}( 1 + T^{4} )^{2} \)
59 \( ( 1 + T^{8} )^{2} \)
61 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
71 \( 1 + T^{16} \)
73 \( ( 1 + T )^{8}( 1 + T^{8} ) \)
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

−5.06239620894146776061362063078, −4.70096464958984022162231711454, −4.57808741067888645370820890259, −4.51072493771622239461044471722, −4.48998568235527809037970386754, −4.35858189279909206451949425978, −4.21763068288587480902041673157, −4.20370402316921004419460849613, −4.10520438691444965839565846077, −3.53372728561153302885629137637, −3.37381165218340105960357164541, −3.34465808366842047067278156145, −3.32720565535012117983580849806, −3.08755474479445968235233227101, −2.94817295402470425033788728748, −2.87912658523284310481614037251, −2.80737861633896188706736572081, −2.53941262515402623269061681640, −2.07373832743468992303578357588, −1.91911946694347466477349107245, −1.75012794395966607867260784287, −1.73852463777248832295259786461, −1.48812981451970089534642670616, −1.36128957131794691255204997358, −0.78114528332852071208586367621, 0.78114528332852071208586367621, 1.36128957131794691255204997358, 1.48812981451970089534642670616, 1.73852463777248832295259786461, 1.75012794395966607867260784287, 1.91911946694347466477349107245, 2.07373832743468992303578357588, 2.53941262515402623269061681640, 2.80737861633896188706736572081, 2.87912658523284310481614037251, 2.94817295402470425033788728748, 3.08755474479445968235233227101, 3.32720565535012117983580849806, 3.34465808366842047067278156145, 3.37381165218340105960357164541, 3.53372728561153302885629137637, 4.10520438691444965839565846077, 4.20370402316921004419460849613, 4.21763068288587480902041673157, 4.35858189279909206451949425978, 4.48998568235527809037970386754, 4.51072493771622239461044471722, 4.57808741067888645370820890259, 4.70096464958984022162231711454, 5.06239620894146776061362063078

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

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