Properties

Degree 16
Conductor $ 3^{16} \cdot 11^{8} $
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  − 2-s + 5·3-s − 4-s − 4·5-s − 5·6-s − 7-s + 10·9-s + 4·10-s − 4·11-s − 5·12-s − 7·13-s + 14-s − 20·15-s + 3·16-s − 10·17-s − 10·18-s + 18·19-s + 4·20-s − 5·21-s + 4·22-s − 14·23-s + 11·25-s + 7·26-s + 10·27-s + 28-s + 6·29-s + 20·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 2.88·3-s − 1/2·4-s − 1.78·5-s − 2.04·6-s − 0.377·7-s + 10/3·9-s + 1.26·10-s − 1.20·11-s − 1.44·12-s − 1.94·13-s + 0.267·14-s − 5.16·15-s + 3/4·16-s − 2.42·17-s − 2.35·18-s + 4.12·19-s + 0.894·20-s − 1.09·21-s + 0.852·22-s − 2.91·23-s + 11/5·25-s + 1.37·26-s + 1.92·27-s + 0.188·28-s + 1.11·29-s + 3.65·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{16} \cdot 11^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{99} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 3^{16} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(0.608575\)
\(L(\frac12)\)  \(\approx\)  \(0.608575\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad3 \( 1 - 5 T + 5 p T^{2} - 35 T^{3} + 67 T^{4} - 35 p T^{5} + 5 p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
11 \( ( 1 + T + T^{2} )^{4} \)
good2 \( 1 + T + p T^{2} + 3 T^{3} + p T^{4} - p^{2} T^{5} + T^{6} - 3 p T^{7} - 11 T^{8} - 3 p^{2} T^{9} + p^{2} T^{10} - p^{5} T^{11} + p^{5} T^{12} + 3 p^{5} T^{13} + p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
5 \( 1 + 4 T + p T^{2} - 18 T^{3} - 94 T^{4} - 232 T^{5} - 22 p T^{6} + 1011 T^{7} + 3826 T^{8} + 1011 p T^{9} - 22 p^{3} T^{10} - 232 p^{3} T^{11} - 94 p^{4} T^{12} - 18 p^{5} T^{13} + p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 + T - 3 p T^{2} - 10 T^{3} + 254 T^{4} + 36 T^{5} - 344 p T^{6} - 83 T^{7} + 18573 T^{8} - 83 p T^{9} - 344 p^{3} T^{10} + 36 p^{3} T^{11} + 254 p^{4} T^{12} - 10 p^{5} T^{13} - 3 p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 7 T + 12 T^{2} + 23 T^{3} + 77 T^{4} - 48 p T^{5} - 1550 T^{6} + 622 T^{7} - 8472 T^{8} + 622 p T^{9} - 1550 p^{2} T^{10} - 48 p^{4} T^{11} + 77 p^{4} T^{12} + 23 p^{5} T^{13} + 12 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
17 \( ( 1 + 5 T + 44 T^{2} + 86 T^{3} + 682 T^{4} + 86 p T^{5} + 44 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 9 T + 4 p T^{2} - 432 T^{3} + 2112 T^{4} - 432 p T^{5} + 4 p^{3} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 14 T + 53 T^{2} + 96 T^{3} + 1775 T^{4} + 8902 T^{5} - 9839 T^{6} + 41958 T^{7} + 1213519 T^{8} + 41958 p T^{9} - 9839 p^{2} T^{10} + 8902 p^{3} T^{11} + 1775 p^{4} T^{12} + 96 p^{5} T^{13} + 53 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 6 T - 71 T^{2} + 192 T^{3} + 4573 T^{4} - 4224 T^{5} - 186713 T^{6} + 77688 T^{7} + 5509213 T^{8} + 77688 p T^{9} - 186713 p^{2} T^{10} - 4224 p^{3} T^{11} + 4573 p^{4} T^{12} + 192 p^{5} T^{13} - 71 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 2 T - 99 T^{2} + 20 T^{3} + 5978 T^{4} + 2478 T^{5} - 249986 T^{6} - 33695 T^{7} + 8127744 T^{8} - 33695 p T^{9} - 249986 p^{2} T^{10} + 2478 p^{3} T^{11} + 5978 p^{4} T^{12} + 20 p^{5} T^{13} - 99 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
37 \( ( 1 - 3 T + 67 T^{2} - 189 T^{3} + 2277 T^{4} - 189 p T^{5} + 67 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 2 T - 97 T^{2} + 612 T^{3} + 4493 T^{4} - 38218 T^{5} - 22457 T^{6} + 901512 T^{7} - 2262647 T^{8} + 901512 p T^{9} - 22457 p^{2} T^{10} - 38218 p^{3} T^{11} + 4493 p^{4} T^{12} + 612 p^{5} T^{13} - 97 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 21 T + 113 T^{2} - 12 p T^{3} + 15670 T^{4} - 131670 T^{5} + 283958 T^{6} - 4210359 T^{7} + 56707993 T^{8} - 4210359 p T^{9} + 283958 p^{2} T^{10} - 131670 p^{3} T^{11} + 15670 p^{4} T^{12} - 12 p^{6} T^{13} + 113 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 7 T - 124 T^{2} + 759 T^{3} + 11189 T^{4} - 50870 T^{5} - 638711 T^{6} + 948885 T^{7} + 33885262 T^{8} + 948885 p T^{9} - 638711 p^{2} T^{10} - 50870 p^{3} T^{11} + 11189 p^{4} T^{12} + 759 p^{5} T^{13} - 124 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \)
53 \( ( 1 + 6 T + 167 T^{2} + 789 T^{3} + 11961 T^{4} + 789 p T^{5} + 167 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 + 2 T - 211 T^{2} - 180 T^{3} + 27194 T^{4} + 9886 T^{5} - 2406650 T^{6} - 211857 T^{7} + 162461212 T^{8} - 211857 p T^{9} - 2406650 p^{2} T^{10} + 9886 p^{3} T^{11} + 27194 p^{4} T^{12} - 180 p^{5} T^{13} - 211 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 15 T - 97 T^{2} - 990 T^{3} + 28072 T^{4} + 155670 T^{5} - 2089990 T^{6} + 70125 T^{7} + 207502789 T^{8} + 70125 p T^{9} - 2089990 p^{2} T^{10} + 155670 p^{3} T^{11} + 28072 p^{4} T^{12} - 990 p^{5} T^{13} - 97 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 14 T - 51 T^{2} - 500 T^{3} + 11354 T^{4} + 102 p T^{5} - 1143698 T^{6} - 2330227 T^{7} + 43235628 T^{8} - 2330227 p T^{9} - 1143698 p^{2} T^{10} + 102 p^{4} T^{11} + 11354 p^{4} T^{12} - 500 p^{5} T^{13} - 51 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \)
71 \( ( 1 + 3 T + 197 T^{2} + 909 T^{3} + 17697 T^{4} + 909 p T^{5} + 197 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 22 T + 421 T^{2} - 4807 T^{3} + 49780 T^{4} - 4807 p T^{5} + 421 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 + 11 T - 171 T^{2} - 1952 T^{3} + 23168 T^{4} + 216654 T^{5} - 1770938 T^{6} - 6655723 T^{7} + 156847173 T^{8} - 6655723 p T^{9} - 1770938 p^{2} T^{10} + 216654 p^{3} T^{11} + 23168 p^{4} T^{12} - 1952 p^{5} T^{13} - 171 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 18 T - 5 T^{2} - 1620 T^{3} - 2021 T^{4} + 27828 T^{5} - 824807 T^{6} + 4969494 T^{7} + 193494259 T^{8} + 4969494 p T^{9} - 824807 p^{2} T^{10} + 27828 p^{3} T^{11} - 2021 p^{4} T^{12} - 1620 p^{5} T^{13} - 5 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 + 6 T + 224 T^{2} + 1554 T^{3} + 26094 T^{4} + 1554 p T^{5} + 224 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + 26 T + 132 T^{2} - 8 p T^{3} + 16487 T^{4} + 336432 T^{5} + 568456 T^{6} + 7259774 T^{7} + 271106712 T^{8} + 7259774 p T^{9} + 568456 p^{2} T^{10} + 336432 p^{3} T^{11} + 16487 p^{4} T^{12} - 8 p^{6} T^{13} + 132 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.68449967536954070543052611740, −6.58999612750236329830392779345, −6.16206457559506069258825059088, −6.03338941022192488838967716479, −5.89825725737881031304735430165, −5.88593171555966075386645531454, −5.45880356007829472884828902541, −5.30947180548035457557014545678, −5.07094737039958634117753956355, −4.74813069034989260249776342890, −4.67025176343136594905340310929, −4.62671001098699171328357433170, −4.20396939852830988930132799869, −4.13303628190339915877173735442, −3.86843007748427001023293042077, −3.83914380739000482339229521299, −3.38145631767120838025571701785, −3.25578403595796048649443444071, −2.77760752557498646334963294429, −2.75932399467404698268855279009, −2.68795680624850277873737133833, −2.53264539477607638353543442603, −2.37980818027998053299919264539, −1.62424820920486031135675820796, −0.74853076515329875380635694455, 0.74853076515329875380635694455, 1.62424820920486031135675820796, 2.37980818027998053299919264539, 2.53264539477607638353543442603, 2.68795680624850277873737133833, 2.75932399467404698268855279009, 2.77760752557498646334963294429, 3.25578403595796048649443444071, 3.38145631767120838025571701785, 3.83914380739000482339229521299, 3.86843007748427001023293042077, 4.13303628190339915877173735442, 4.20396939852830988930132799869, 4.62671001098699171328357433170, 4.67025176343136594905340310929, 4.74813069034989260249776342890, 5.07094737039958634117753956355, 5.30947180548035457557014545678, 5.45880356007829472884828902541, 5.88593171555966075386645531454, 5.89825725737881031304735430165, 6.03338941022192488838967716479, 6.16206457559506069258825059088, 6.58999612750236329830392779345, 6.68449967536954070543052611740

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

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