Properties

Label 16-12e24-1.1-c3e8-0-7
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $1.16755\times 10^{16}$
Root an. cond. $10.0972$
Motivic weight $3$
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  − 1.22e3·23-s − 496·25-s + 1.80e3·47-s + 1.25e3·49-s − 432·71-s + 344·73-s + 1.24e3·97-s + 3.80e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.31e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 11.0·23-s − 3.96·25-s + 5.58·47-s + 3.65·49-s − 0.722·71-s + 0.551·73-s + 1.29·97-s + 2.86·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 5.98·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(1.16755\times 10^{16}\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(5.310147773\)
\(L(\frac12)\) \(\approx\) \(5.310147773\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 + 248 T^{2} + 1806 p^{2} T^{4} + 248 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
7 \( ( 1 - 626 T^{2} + 327363 T^{4} - 626 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
11 \( ( 1 - 1904 T^{2} + 3668622 T^{4} - 1904 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
13 \( ( 1 - 3287 T^{2} + p^{6} T^{4} )^{4} \)
17 \( ( 1 - 5072 T^{2} + 1982238 T^{4} - 5072 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
19 \( ( 1 + 18094 T^{2} + 171635955 T^{4} + 18094 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
23 \( ( 1 + 306 T + 44422 T^{2} + 306 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
29 \( ( 1 + 41866 T^{2} + p^{6} T^{4} )^{4} \)
31 \( ( 1 - 116204 T^{2} + 5150826150 T^{4} - 116204 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
37 \( ( 1 - 92182 T^{2} + 7044936315 T^{4} - 92182 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
41 \( ( 1 - 30740 T^{2} + 8341944198 T^{4} - 30740 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
43 \( ( 1 + 65524 T^{2} + 6829649142 T^{4} + 65524 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
47 \( ( 1 - 450 T + 95542 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
53 \( ( 1 + 154436 T^{2} + 4597192086 T^{4} + 154436 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
59 \( ( 1 - 512528 T^{2} + 126780934542 T^{4} - 512528 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
61 \( ( 1 - 763582 T^{2} + 246326003187 T^{4} - 763582 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
67 \( ( 1 + 743782 T^{2} + 274440697755 T^{4} + 743782 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
71 \( ( 1 + 108 T + 599182 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
73 \( ( 1 - 86 T + 65499 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
79 \( ( 1 - 1357322 T^{2} + 905419894299 T^{4} - 1357322 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
83 \( ( 1 - 1580396 T^{2} + 1168518522966 T^{4} - 1580396 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
89 \( ( 1 - 2136560 T^{2} + 2018716991358 T^{4} - 2136560 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
97 \( ( 1 - 310 T + 926871 T^{2} - 310 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
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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

−3.66489732101163103783555237401, −3.61288267353500115047653028724, −3.40382516159225852303356863184, −3.14615878993513863534724432936, −2.96752920942207578750289068673, −2.84264331635991471878985127219, −2.66842627000134250803722354669, −2.41811533278654183087506696835, −2.36660935033929543148065666068, −2.35798840525196231921822749968, −2.23194074845127563602157206050, −2.09557074685025815175467306516, −1.95858100693458523988758172688, −1.84134734956585573308492624830, −1.75597054537248759573926919662, −1.70568421921254199419729209298, −1.57143844137436173321259423304, −1.39094836422120140781856550921, −0.941082706432689330108378660275, −0.68745856982774533054979771533, −0.63944405499607470884192613388, −0.61254775024283876305884808094, −0.28291095759954887472252910759, −0.27616604483747964077452382572, −0.24262335833390788997963994625, 0.24262335833390788997963994625, 0.27616604483747964077452382572, 0.28291095759954887472252910759, 0.61254775024283876305884808094, 0.63944405499607470884192613388, 0.68745856982774533054979771533, 0.941082706432689330108378660275, 1.39094836422120140781856550921, 1.57143844137436173321259423304, 1.70568421921254199419729209298, 1.75597054537248759573926919662, 1.84134734956585573308492624830, 1.95858100693458523988758172688, 2.09557074685025815175467306516, 2.23194074845127563602157206050, 2.35798840525196231921822749968, 2.36660935033929543148065666068, 2.41811533278654183087506696835, 2.66842627000134250803722354669, 2.84264331635991471878985127219, 2.96752920942207578750289068673, 3.14615878993513863534724432936, 3.40382516159225852303356863184, 3.61288267353500115047653028724, 3.66489732101163103783555237401

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

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