Properties

Degree $16$
Conductor $7.985\times 10^{28}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·11-s − 12·19-s + 8·29-s − 16·31-s − 8·41-s + 12·49-s − 24·59-s + 48·61-s − 4·71-s + 36·79-s − 4·81-s − 8·89-s − 24·101-s − 40·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2.41·11-s − 2.75·19-s + 1.48·29-s − 2.87·31-s − 1.24·41-s + 12/7·49-s − 3.12·59-s + 6.14·61-s − 0.474·71-s + 4.05·79-s − 4/9·81-s − 0.847·89-s − 2.38·101-s − 3.83·109-s − 0.363·121-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 + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{16} \cdot 41^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{4100} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{16} \cdot 41^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5404407845\)
\(L(\frac12)\) \(\approx\) \(0.5404407845\)
\(L(\frac{3}{2})\) 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 \( 1 \)
41 \( ( 1 + T )^{8} \)
good3 \( 1 + 4 T^{4} - 4 p^{2} T^{6} - 14 T^{8} - 4 p^{4} T^{10} + 4 p^{4} T^{12} + p^{8} T^{16} \)
7 \( 1 - 12 T^{2} + 12 p T^{4} - 200 T^{6} - 558 T^{8} - 200 p^{2} T^{10} + 12 p^{5} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 - 4 T + 26 T^{2} - 114 T^{3} + 384 T^{4} - 114 p T^{5} + 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 24 T^{2} + 380 T^{4} - 4584 T^{6} + 59814 T^{8} - 4584 p^{2} T^{10} + 380 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 24 T^{2} + 28 p T^{4} - 40 p T^{6} + 3398 T^{8} - 40 p^{3} T^{10} + 28 p^{5} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 6 T + 62 T^{2} + 208 T^{3} + 1448 T^{4} + 208 p T^{5} + 62 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 72 T^{2} + 100 p T^{4} - 40952 T^{6} + 679622 T^{8} - 40952 p^{2} T^{10} + 100 p^{5} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 4 T + 76 T^{2} - 12 p T^{3} + 2870 T^{4} - 12 p^{2} T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 8 T + 92 T^{2} + 712 T^{3} + 3846 T^{4} + 712 p T^{5} + 92 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 168 T^{2} + 14516 T^{4} - 846120 T^{6} + 36244134 T^{8} - 846120 p^{2} T^{10} + 14516 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 232 T^{2} + 26780 T^{4} - 1971224 T^{6} + 100629414 T^{8} - 1971224 p^{2} T^{10} + 26780 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 216 T^{2} + 25412 T^{4} - 1955324 T^{6} + 108019538 T^{8} - 1955324 p^{2} T^{10} + 25412 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 168 T^{2} + 17692 T^{4} - 1367128 T^{6} + 81443238 T^{8} - 1367128 p^{2} T^{10} + 17692 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 12 T + 252 T^{2} + 1996 T^{3} + 22582 T^{4} + 1996 p T^{5} + 252 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 24 T + 420 T^{2} - 4824 T^{3} + 44086 T^{4} - 4824 p T^{5} + 420 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 292 T^{2} + 46132 T^{4} - 4949152 T^{6} + 385553458 T^{8} - 4949152 p^{2} T^{10} + 46132 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 2 T + 98 T^{2} - 268 T^{3} + 48 p T^{4} - 268 p T^{5} + 98 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 360 T^{2} + 67764 T^{4} - 8331176 T^{6} + 717581958 T^{8} - 8331176 p^{2} T^{10} + 67764 p^{4} T^{12} - 360 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 18 T + 366 T^{2} - 4224 T^{3} + 45328 T^{4} - 4224 p T^{5} + 366 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 360 T^{2} + 73244 T^{4} - 9891864 T^{6} + 961607526 T^{8} - 9891864 p^{2} T^{10} + 73244 p^{4} T^{12} - 360 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 4 T + 228 T^{2} + 796 T^{3} + 326 p T^{4} + 796 p T^{5} + 228 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 280 T^{2} + 40028 T^{4} - 3597992 T^{6} + 303244998 T^{8} - 3597992 p^{2} T^{10} + 40028 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.55286021176547737317131479639, −3.43675698262255137316389845298, −3.22703048381050470202608454077, −3.14390690256427304865427339336, −2.98717680689894999762254306896, −2.82043650841908994701810052938, −2.78493096707027308487155970811, −2.72141268950530841983903766108, −2.46173986802052907594294309606, −2.43092277511463363102581008465, −2.16691143668347396865640440920, −2.14316691640828768757521265591, −2.06970770955272806239507509046, −1.82520777247152123741553858143, −1.81280879084086269723504677444, −1.68054963858447922746781576200, −1.67108024071376353396434373532, −1.23257228995306367619353025466, −1.19226306949221549833061345847, −1.10682322114801809465555024742, −0.925453176964695748369816735845, −0.74489697905913696173078571311, −0.62605776915274697799387293831, −0.19601820553194928549736211281, −0.079723207578526011100499935300, 0.079723207578526011100499935300, 0.19601820553194928549736211281, 0.62605776915274697799387293831, 0.74489697905913696173078571311, 0.925453176964695748369816735845, 1.10682322114801809465555024742, 1.19226306949221549833061345847, 1.23257228995306367619353025466, 1.67108024071376353396434373532, 1.68054963858447922746781576200, 1.81280879084086269723504677444, 1.82520777247152123741553858143, 2.06970770955272806239507509046, 2.14316691640828768757521265591, 2.16691143668347396865640440920, 2.43092277511463363102581008465, 2.46173986802052907594294309606, 2.72141268950530841983903766108, 2.78493096707027308487155970811, 2.82043650841908994701810052938, 2.98717680689894999762254306896, 3.14390690256427304865427339336, 3.22703048381050470202608454077, 3.43675698262255137316389845298, 3.55286021176547737317131479639

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

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