Properties

Label 16-288e8-1.1-c1e8-0-0
Degree $16$
Conductor $4.733\times 10^{19}$
Sign $1$
Analytic cond. $782.270$
Root an. cond. $1.51647$
Motivic weight $1$
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  + 2·5-s − 3·9-s − 6·13-s + 4·17-s + 5·25-s + 10·29-s + 32·37-s − 4·41-s − 6·45-s + 49-s − 32·53-s − 26·61-s − 12·65-s + 60·73-s − 3·81-s + 8·85-s + 56·89-s − 36·97-s + 26·101-s + 48·109-s − 58·113-s + 18·117-s + 8·121-s − 6·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 9-s − 1.66·13-s + 0.970·17-s + 25-s + 1.85·29-s + 5.26·37-s − 0.624·41-s − 0.894·45-s + 1/7·49-s − 4.39·53-s − 3.32·61-s − 1.48·65-s + 7.02·73-s − 1/3·81-s + 0.867·85-s + 5.93·89-s − 3.65·97-s + 2.58·101-s + 4.59·109-s − 5.45·113-s + 1.66·117-s + 8/11·121-s − 0.536·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(782.270\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{288} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.131274562\)
\(L(\frac12)\) \(\approx\) \(3.131274562\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p T^{2} + 4 p T^{4} + p^{3} T^{6} + p^{4} T^{8} \)
good5 \( ( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 8 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \)
7 \( 1 - T^{2} - 23 T^{4} + 74 T^{6} - 1874 T^{8} + 74 p^{2} T^{10} - 23 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 8 T^{2} + 103 T^{4} + 2248 T^{6} - 20864 T^{8} + 2248 p^{2} T^{10} + 103 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 + 3 T - 11 T^{2} - 18 T^{3} + 114 T^{4} - 18 p T^{5} - 11 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - T + 26 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \)
19 \( ( 1 + 31 T^{2} + 888 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 - 65 T^{2} + 95 p T^{4} - 63830 T^{6} + 1646734 T^{8} - 63830 p^{2} T^{10} + 95 p^{5} T^{12} - 65 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 5 T - 31 T^{2} + 10 T^{3} + 1570 T^{4} + 10 p T^{5} - 31 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 11 T^{2} - 1163 T^{4} - 7018 T^{6} + 608854 T^{8} - 7018 p^{2} T^{10} - 1163 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} \)
37 \( ( 1 - 4 T + p T^{2} )^{8} \)
41 \( ( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \)
43 \( 1 - 136 T^{2} + 10471 T^{4} - 588472 T^{6} + 26782720 T^{8} - 588472 p^{2} T^{10} + 10471 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 161 T^{2} + 15097 T^{4} - 1031366 T^{6} + 55019806 T^{8} - 1031366 p^{2} T^{10} + 15097 p^{4} T^{12} - 161 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 + 4 T + p T^{2} )^{8} \)
59 \( 1 - 56 T^{2} - 4313 T^{4} - 27272 T^{6} + 32453824 T^{8} - 27272 p^{2} T^{10} - 4313 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 + 13 T + 79 T^{2} - 416 T^{3} - 5930 T^{4} - 416 p T^{5} + 79 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 88 T^{2} - 2873 T^{4} - 144232 T^{6} + 57806752 T^{8} - 144232 p^{2} T^{10} - 2873 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 176 T^{2} + 16638 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 15 T + 194 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( 1 - 181 T^{2} + 12757 T^{4} - 1361482 T^{6} + 156748534 T^{8} - 1361482 p^{2} T^{10} + 12757 p^{4} T^{12} - 181 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 125 T^{2} + 6925 T^{4} + 634750 T^{6} - 79408946 T^{8} + 634750 p^{2} T^{10} + 6925 p^{4} T^{12} - 125 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 14 T + 194 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \)
97 \( ( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \)
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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

−5.32192992182478983576924522287, −5.31530258382124925060140407248, −4.86437345975321369998086933843, −4.78672980226981932131700552568, −4.73572564315703452396282763415, −4.56230258661595443503977275590, −4.48147036015176642210373706610, −4.42525794540025393337686531930, −4.35957908540810196214737763617, −3.74494744379486701084903876311, −3.67627620266510496567358777242, −3.40274227448249267615359237043, −3.37736341442977385034997347893, −3.27044708014970140269218717457, −2.95163567511660650148391584590, −2.87360208976483876256605593429, −2.56067843372013027783040295855, −2.50129024771432714440626709300, −2.23110178459491883231534488872, −2.20249323023959182286148971080, −1.85660372514842916522526418044, −1.52413947575859818087780277378, −1.15561251625060070634372675527, −0.844747079800542815861732081025, −0.56742523273415704132844342604, 0.56742523273415704132844342604, 0.844747079800542815861732081025, 1.15561251625060070634372675527, 1.52413947575859818087780277378, 1.85660372514842916522526418044, 2.20249323023959182286148971080, 2.23110178459491883231534488872, 2.50129024771432714440626709300, 2.56067843372013027783040295855, 2.87360208976483876256605593429, 2.95163567511660650148391584590, 3.27044708014970140269218717457, 3.37736341442977385034997347893, 3.40274227448249267615359237043, 3.67627620266510496567358777242, 3.74494744379486701084903876311, 4.35957908540810196214737763617, 4.42525794540025393337686531930, 4.48147036015176642210373706610, 4.56230258661595443503977275590, 4.73572564315703452396282763415, 4.78672980226981932131700552568, 4.86437345975321369998086933843, 5.31530258382124925060140407248, 5.32192992182478983576924522287

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

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